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Lucy Frazer – Tory MP – Destined for Greatness – Jewess – Strangely Seemingly an Admirer of Perpetrators of Shoahs – But Only if Scots Are on the Receiving End of the Brutality
Lucy Frazer (1972), is a Conservative Party politician.
Born and raised in Leeds, Yorkshire. She is the descendant of Russian Jewish immigrants. (Frazer is a common Jewish surname)
Her paternal Great Grandfather was Jewish immigrant: Barnet Frazer. ( A tailor to trade, Russian born (1880-1933) settled in Leeds (married Elizabeth (1870-1925) in 1897.
Her maternal Great Grandfather was Jewish immigrant: Soloman Leviten. (Russian born (1870 – 1923) settled in Leeds (married Elizabeth Perlston (1875-1940) in 1898.
Her paternal Grandfather was Hyman Frazer (1908-1985) (mathematician, barrister and educationalist, Headmaster of Gateway College in Leicester, married in Leeds in 1933.)
Her maternal Grandmother was Yetta Leviten (1913-1995) (Last born of 8 children)
Her father, Colin Peter Frazer, was a partner in the collapsed Leeds legal firm Fox Hayes. (see below)
Lucy was educated at Leeds Girls' High School, an independent school in Headingley, a suburb of the City of Leeds, followed by Newnham College, Cambridge, where she was President of the Cambridge Union. She worked as a barrister in commercial law specialising in major insolvency and restructuring cases, including being a member of the legal team in the Fox Hayes v. Financial Services Authority court case in 2010, and went on to become a QC in 2013 at the age of forty.
She is married to millionaire businessman, David Bernard Leigh, (Levy?) Possessing an MA in Social and Political Sciences from Cambridge University and a post-graduate legal qualification from the College of Law, London he is a Director of a number of financially successful concerns.
Lucy successfully stood (after a bit of controversy. See below) for election to Westminster in 2015 and is the current Member of Parliament (MP) for South East Cambridgeshire.
2011: Lucy Frazer's Father Guilty of "Boiler Room" Fraud Promoting Unauthorised Financial Promotions
The Financial Services Authority (FSA) has found the Leeds-based law firm Fox Hayes partners to be personally liable for breaches of the "Conduct of Business" (COB) in relation to the approval of at least 20 financial promotions for five overseas unauthorised entities between February 2003 and June 2004. UK customers (670) purchased shares totalling approximately $20 million which are now of little or no value.
The FSA imposed a fine of £454,770 on the now defunct Fox Hayes partnership: comprising Mr John Robert Manning, Mr Stephen Maurice Coupland, Mr Ian David Coupland, Mr Philip Leigh Drazen, Mr Malcolm Richard Jones, Mr Richard Geraint Jones, Mr Colin Peter Frazer and Mr Ian Brill.
2014: Two international share swindlers are starting lengthy jail sentences after being convicted in the biggest 'boiler room' investment case ever brought by the Serious Fraud Office.
Australian businessman Jeffrey Revell-Reade, 49, was sentenced on Friday to eight and a half years in prison, and his British accomplice Anthony May, 58, was jailed for seven years and four months after they were found guilty at South London's Southwark Crown Court of conspiracy to defraud. The two men controlled a string of bogus stockbroking firms based in Spain that used false claims and high-pressure phone calls to persuade more than 1,000 investors to part with about £70million for shares that proved worthless.
A third man, Robert Manning, 63, formerly senior partner in Leeds solicitors Fox Hayes, was accused of pocketing several hundred thousand pounds in return for lending his firm's name to promotional mailshots issued by the brokers and collecting payments from investors, but the jury cleared him on a charge of corruption.
Daniel Gooding, 39, of Brentwood, Essex, was sentenced to seven years after the court heard he was the manager who ran the telemarketing sales teams in Spain. Shaun Rumsey, 33 of Basildon, Essex, Philip Morris, 38, of Biddenden, Kent, and Jon Emery, 36, of Romford, Essex, were each jailed for five years.
Ian Hughes, 34, of Hainault, Essex, was sentenced to four years. Gooding's sister Emma Farmer, 41, was jailed for three years. Her mother, Michelle Coleman, 62, was given a 21-month sentence, suspended for two years, after she admitted laundering £252,425, of which she kept £158,000.
The Serious Fraud Office investigation – codenamed Operation Steamroller – lasted seven years and involved enquiries in the US, Hong Kong and New Zealand. Revell-Reade was extradited from Australia in 2012, and May was arrested in Spain several months later.
Revell-Reade is believed to have made about £37million from the frauds. He bought three apartments in Marbella, Spain, and a £5million luxury home in Wimbledon, South-West London, as well as hiring private jets and spending £13,000 on Rolex and Cartier watches.
July 2003: Tresaderns & Partners, based in Madrid, was linked to an earlier boiler room scam, and Spanish company records showed it was posing as a credit checking business, not a stockbroker. The information was made known to solicitors Fox Hayes, together with the warning that it was putting its own reputation on the line in fronting for crooks. What was not known at the time was that, unknown to his partners, the law firm's boss Mr John Robert Manning had done a secret deal with Revell-Reade that earned him 4 per cent of all the cash raised from victims.
In 2004, I investigated Atlantic Wine Agencies, an Australian vineyard company whose shares were marketed by Tresaderns and by the fully licensed – but disreputable – London broker Pacific Continental Securities. I named Revell-Reade as one of those behind the scam.
Much later, the Financial Services Authority found evidence that he virtually controlled the London firm behind the scenes. It was heavily fined and went bust in 2007. The fraud charges against the share crooks also involved two other offshore brokers, Price Stone and Anderson McCormack.
Evidence was given that Revell-Reade had held meetings with Fox Hayes to discuss how to respond to MoS revelations. And when too many awkward questions were raised, one firm would disappear and a new name would take over. I warned against the two new boiler rooms in 2006 and named Daniel Gooding as a main player.
But just as interesting as those in the dock are the names that cropped up in evidence but faced no charges. One man deeply involved in dealing with Fox Hayes was David Rycott. More than 20 years ago, he ran a rip-off commodity investment firm in London called DPR Futures. He was cleared of fraud charges at the Old Bailey and went on to run an unlicensed foreign exchange dealership before leaving the country.
Another was London lawyer Andrew Greystoke. He threatened to sue me when I reported in 2009 that his firm Atlantic Law had been fined for its links to Anderson McCormack and three other boiler rooms.
The case against Revell-Reade, May and their sales manager, Gooding, revolved around three scams, but there were more: Hoffman Philips, City Allied, Eagle Star International and others. They even took over a Berlin broking firm, Atlantic Capital, that was shut down after it was raided by police in 2007.
https://www.fca.org.uk/publication/final-notices/fox_hayes.pdf
http://www.thisismoney.co.uk/money/experts/article-2651479/TONY-HETHERINGTON-Jail-70m-fraudsters-exposed-Mail-Sunday.html
http://redcatsboards.yuku.com/topic/34437
David Bernard Leigh
David Bernard Leigh (Levy) and Borderless Higher Education
Over the next decade it is anticipated upwards of 10 million students will be seeking university placements outside the country of their birth so that they will be able to gain enabling qualifications assuring a successful career in an ever increasing international market. The uncompetitive UK higher education system is not yet fully adjusted to the new reality.
Government and University authorities have been slow in recognising that the number of international students coming to Europe, embracing for-profit institutions is increasing steadily and a smaller number of UK based students are expressing interest in completing their first year of study enrolling with a for-profits institution (with lower fees than public funded universities.)
The progression of borderless education has introduced to the UK, the for profit, "Pathway Centre" meeting demands of an ever increasing number of international students. The emergence of these for-profit universities in England did attract comment in the national media, but the rapid growth of private study centres offering international students pre-degree "pathway" courses into higher education has remained below the radar. Yet thousands of international students now study at these for profit centres which offer courses linked to a guaranteed place on a degree course at a partner university.
The global leader the "Study Group" has, as it's Chief Executive Officer (CEO) David Bernard Leigh (Levy?) The Group operates its own colleges, preparing students for international academic success in collaboration with leading universities across the UK, Europe, North America and Australasia providing on and off-campus education through International Study Centres (ISCs). It's global recruitment teams enrolled over 73,000 students from 145 countries in 2015.
And a number of UK universities have embraced the Pathway concept including, the "Northern Consortium" institution, (comprising the universities of Bradford, Huddersfield, Leeds, Leeds Metropolitan, Liverpool, Liverpool John Moores, Manchester, Manchester Metropolitan, Salford, Sheffield and Sheffield Hallam.)
https://www.timeshighereducation.com/features/pathways-to-profit/2012075.article https://www.facebook.com/StudyGroupCorporate/
Wife: Lucy Frazer MP, got herself elected onto the House of Commons, Education Select Committee. Hows that for blatant nepotism?
David Bernard Leigh also retains directorships with:
*The Hall School Charitable Trust. A fee paying private school for boys.
* The London Carlbach Shul Trust. Objectives: Advancement of the orthodox Jewish religion by the establishment continuance and support of an orthodox Jewish synagogue.
* International Entertainment Holdings Limited. Worlds largest performing arts group, operating theatres, producing, theatrical productions. Greg Dykes Chairman. Value £600million.
* EDU-UK Bondco PlC. Holding company and 3 subsidiaries.
More details at: https://beta.companieshouse.gov.uk/officers/1-n1EQvhyu0ogjPPac3l3Z3p3S0/appointments
21 January 2014: Resignations follow Cambridgeshire Tory vote 'shambles'
Members of the Conservative Party in Cambridgeshire have resigned over concerns about the selection of a general election candidate. Lucy Frazer had been selected by David Cameron to stand for selection in a primary for the safe seat of South East Cambridgeshire. The local party looked at the election again after allegations votes had been miscounted, but chose not to rerun it. The votes actually put another candidate in the lead.
At an emergency meeting of the local Conservative Party a majority of members voted to keep her as the candidate. Peter Johnson, Conservative councillor (who later resigned) said "If there is any doubt with an election, you should rerun it. We have become a laughing stock. We continue to compound that by trying to justify the decision. It is totally untenable. There are others I know who have resigned who are party workers. They work for the party at election time. They have gone and I don't know how the party is going to replace them."
http://www.wisbechstandard.co.uk/news/ballot_gate_farmer_quits_se_cambs_tories_after_group_ratifies_lucy_as_2015_election_candidate_1_3214637
2015: Lucy Frazer's Sharon Stone Moment in the House of Commons
She flashed more than she bargained for in a short skirt at yesterday's PMQs. She sent social media into a frenzy when she sat behind Prime Minister Theresa May and appeared to copy the seductive pose made famous in the 1990 film Basic Instinct.
https://www.thesun.co.uk/news/1747264/tory-mp-lucy-frazer-sparks-twitter-storm-over-her-sharon-stone-moment-during-pmqs/
2015: Lucy Frazer Praises Cromwell's Purge of Scots after the Battle of Dunbar
In her maiden speech in the Commons at Westminster, Lucy Frazer, MP for Cambridgeshire South East, praised Oliver Cromwell, (who was born in her constituency) for his treatment of the Scots Covenanters after the "Battle of Dunbar" when he despatched over 5,000 captive soldiers into slavery. Amid much laughter on the Tory benches she went on to offer her colleagues a view that the answer to the "West Lothian Question", might be to follow Cromwell's lead and banish troublesome Scots to the colonies as slaves.
The speaker was invited, by public petition to rule that she should make a statement to the House apologising for the remarks which had caused great offence to Scots viewers of the Parliamentary Channel and many other UK residents He refused to intervene stating that she was free to say anything that she wished to in Parliament.
His support of Ms Frazer needs to be set against his rebuke of Scottish MP's only a few days earlier when he told them that "clapping" was not allowed in the Commons. Clarification of the rules of the game regarding Scots. It is perfectly acceptable to propose that troublesome Scots MP's should be clapped in irons to be sold as slaves and shipped to the colonies, but Honourable Members must not support the proposal by clapping.
Addressing the banality of her speech it might have been possible to dismiss it (and the conduct of her colleagues) had her comments been "off the cuff" but they had been carefully crafted, written and rehearsed beforehand indicating an undefined level of complicity revealing the patronising ambivalent attitude (towards Scots) of Unionist MP's at Westminster.
That she felt able to make fun of so many young Scots, murdered and enslaved in only one of many Holocausts inflicted by the English on Scotland over many centuries is even more sickening when her own family background is revealed.
The Battle of Dunbar
Following defeat in the English Civil War, Charles I was executed in January 1649. After declaring the establishment of a Commonwealth in England Cromwell took his "New Model Army" over to Ireland and quelled any uprising in a number of victories over Irish supporters of the monarchy.
Charles II was forced to abandon his plans to use Ireland as a military base to win back the throne of England and turned his attention to Scotland, taking advantage of Scottish outrage at the presumption of the English Parliament in executing Charles I, who had been King of Scotland as well as England. Negotiations between Charles and a delegation from the Scottish government opened in the Netherlands in March 1650 which resulted in the signing of the "Treaty of Breda" in May and Charles' arrival in Scotland in June 1650, where he was proclaimed King Charles II of Scotland
The Commonwealth Council of State in London was alarmed when the treaty was signed and resolved to mount an immediate invasion of Scotland to forestall the possibility of a Scottish attack on England and in July the English Parliament, expecting Charles to initiate a Scottish led campaign for the English crown, launched a pre-emptive invasion of Scotland.
Oliver Cromwell was appointed commander-in-chief. His battle hardened army comprised seven regiments of horse, nine of foot and six companies of dragoons, numbering around 15,000 men in total. Marching north via York, Durham and Newcastle, Cromwell mustered his forces at Berwick-on-Tweed on 19 July 1650. He crossed the border on 22 July 1650. Crucially, the army was supported by a large supply fleet, which landed at Dunbar a few days before the battle.
The Scottish Parliament belatedly became aware of English preparations for an invasion and, on 25 June 1650, ordered that a new force comprising around 10,000 foot and 3,000 horse be recruited and trained to reinforce the Army of the Covenant which numbered around 1500 foot soldiers and a small cavalry force.
Whilst his ageing father the Earl of Leven took overall command, his son, Lieutenant-General David Leslie son of the Earl of Leven undertook day to day control of the Scottish army (reporting to a Kirk commission headed by Sir Archibald Johnston.) The commissioners insisted that drill or military operations would not be carried out on the Sabbath and attempted to purge the army of all Engagers and potential Royalist supporters and, as the campaign developed, the commissioners interfered in military decisions regarding strategy and tactics.
The summer of 1650 was extremely cold and wet and the Scottish army, (largely comprised of raw recruits aged between 12-18) was still mustering when Cromwell crossed the border from Berwick on 22 July 1650.
Playing for time (to train his young soldiers) Lord Leven directed the construction of an extensive line of earthworks between Edinburgh and Leith which allowed the Scottish army to take up an impregnable defensive position. He also ordered the destruction of all crops and the removal of all livestock between Edinburgh and the border so that Cromwell's army would have to get all its supplies from England.
The English army advanced to within a few miles of Edinburgh but Cromwell realised that the Scottish lines of defence were too strong to risk a direct assault and retreated to the port of Dunbar so that he could occupy its sheltered harbour, securing a sea-route for supplies. Then, under cover of his warships bombarding Leith, Cromwell deployed his troops in battle order in an attempt to draw the Scottish army out into the open.
The Covenanter committee were keen to attack the English force but the seasoned soldier, Lord Leven was equally determined to avoid a pitched battle with his inexperienced army and stayed stubbornly dug in behind the defences.
It rained heavily all week and Cromwell was forced to withdraw, (harassed by Scottish Lancers) first to Musselburgh then to Dunbar for shelter. His army was exhausted, many were sick from cold and flu and food rationing was severe since the supply ships had given priority to the carriage of munitions rather than food. The stand-off lasted throughout August.
The Scottish army had increased in number over the period, swelling to around 20,000 and Lieutenant-General David Leslie Sir Leslie, seized upon the opportunity to move his force around Cromwell cutting off his escape route south to Berwick. Cromwell now finally had Leslie offering battle, but his "New Model Army" was at a severe disadvantage. Cromwell was forced to meet the challenge, achieving what was arguably the most dramatic victory of the Civil Wars.
On 1 September 1650, Leslie took up a commanding position overlooking the English encampment at Dunbar. Trapped between the sea on one side, the Covenanters' impregnable position on the other, and with the road back to England blockaded, it seemed that Cromwell had no option but to attempt to evacuate his troops by sea.
On the morning of 2 September, however, the Covenanters played into Cromwell's hands by marching down from the commanding heights of Doon Hill. The decision to come down from the hill to attack the English army on level ground has traditionally been ascribed to the godly committee that accompanied the army, and is widely condemned as a tactical blunder.
However, the strength of the Scottish position precluded any possibility that Cromwell would attack up the steep slopes of Doon Hill. Lacking heavy artillery, the Scots could do nothing to hinder the English army. Furthermore, the bad weather was worsening and the Scottish army had no shelter from the wind and rain on the exposed hilltop.
During the morning and afternoon of 2 September, the Covenanter army moved down from the hill and drew up in an arc aligned with the course of a stream called the Broxburn, with the coast on the right flank. On the left of the Scottish position, the Broxburn passed through a deep ravine; on the right, towards the coast, the ground levelled out and the stream was crossed by the road from Dunbar to Berwick.
While the Scottish army re-deployed, Cromwell brought the English army forward from Dunbar to form a battle-line on the northern side of the Broxburn. Skirmishing broke out during the late afternoon of 2 September when an advance guard of lancers moved forward to cover the cumbersome manoeuvring as the Scots moved into their new positions.
By nightfall, the Berwick road was successfully blocked. Cromwell was left with a choice between evacuating his army by sea or forcing a way through the Scottish blockade to escape by land. However, while observing the new Scottish position, Cromwell and Lambert both identified a tactical flaw in the deployment.
The Scottish line was awkwardly positioned between the Broxburn river and the slopes of Doon Hill, leaving the centre and left flank with little room for manoeuvre.
At a council of war that night, the English officers agreed to mount an all-out assault on the Scottish right flank early the next morning, with the objective of turning the flank and disrupting the whole position by driving the Scots into the constricted centre. During the night, while the Scots rested uncomfortably in rain-soaked fields, Cromwell used the darkness and heavy rain to cover the redeployment of his army.
The battle line along the Broxburn was abandoned and the English army stacked up, one brigade behind another, across the Berwick road. Around 7,500 infantry and 3,500 cavalry were fit to fight on the morning of 3 September.
The leading vanguard comprised six cavalry regiments comprising 9,000 men and an infantry brigade of around 2,000 men. The vanguard was supported by two more infantry brigades of around 2,500 men each. Cromwell's own regiment of horse were held in reserve, brigaded with two companies of dragoons. The remaining dragoons mounted guard along the edge of the Broxburn to cover the English artillery, which was deployed on rising ground overlooking the Scottish left wing.
The disposition of the Scottish army is not exactly known, but most of the cavalry (numbering around 2,500 troopers) were stationed on the right wing between the Berwick road and the coast, facing the 16,000 strong English vanguard. A cavalry force, numbering around 1,500 was stationed on the left wing. Between the wings of cavalry, there were five brigades of infantry numbering around 10,000.
The Battle of Dunbar began at about four o'clock on the morning of 3 September 1650, when the English vanguard, advanced in force securing the crossings of the Broxburn on the Scottish right flank. Scottish piquets were driven back but raised the alarm. An infantry brigade advanced in support the English cavalry, a fierce firefight and artillery exchange ensued. After nearly an hour, both sides stopped firing and waited for first light, which came at about 5.30 a.m.
Many of the senior Scottish officers had left their units during the night to seek shelter from the foul weather and by result the Scottish army had not fully regrouped by the time the English attack resumed at first light. The attack, spearheaded by a cavalry regiment advanced swiftly across the Broxburn and routed the first line of the Scottish Lancers on the Scottish right flank but the second line of Scottish Lancers fought back and forced the English cavalry to retreat back across the river.
Meantime an English infantry vanguard of 3,000 advanced across the Broxburn and attacked a Scottish brigade of 2,000 (made up almost entirely of raw recruits) on the extreme right of the Scottish infantry line. The Scottish line buckled but a second brigade first halted the English advance then forced it to retreat back across the river. Denied an early breakthrough Cromwell immediately ordered another infantry brigade into action in support of the retreating infantry force. The fall-back was reversed and the Scottish infantry brigades were engaged in a "push of pike" battle.
Cromwell then hit back deploying two regiments of cavalry 3,000 to attack the Scottish cavalry, (reduced through the earlier attrition to around 1,000.) The Scottish cavalry was put to flight leaving the infantry on the right flank of the Scottish army exposed. Cromwell seized the opportunity and ordered his own Regiment of Horse into battle attacking the exposed Scottish right flank
The routing of the Scottish cavalry opened the possibility of a decisive victory over the Scottish army, rather than simply breaking the blockade of the road back to England and Cromwell ordered his cavalry to attack the exposed Scottish infantry right flank. This forced the Scots infantry to fight on two fronts and pushed them into the forces to their left reducing their ability to manoeuvre effectively. After a desperate battle lasting two hours the heavily outnumbered Scots on the right flank were forced to fight a protracted withdrawal eventually conceding defeat in the village of Haddington. Scots forces on the left flank abandoned the battle and escaped to the North. Cromwell went onto ruthlessly ransack Edinburgh and other Scottish towns and cities and take control of the country south of the Highlands.
descendants of Scottish soldiers sold as slaves by Cromwell after the battle of Dunbar
The battle left nearly 5,000 Scots dead. 6,000 Scottish soldiers (aged between 13 and 25) were taken prisoner. Of these about 1,000 sick and wounded were released to go home. The remaining 5,000 battle weary Scots prisoners were forced to complete a brutal 8-day, 118 miles forced march south to the English city of Durham with virtually no rest (the first 28 mile stage to Berwick being undertaken non-stop through the night) and with no food or water, other than what could be scavenged. About 1,000 prisoners died en route to Durham from a combination of hunger, exhaustion and probably dysentery. Others were executed and some escaped.
Around 3,000 Scottish soldiers were subsequently imprisoned in Durham Cathedral and Castle, (unoccupied since worship had been suppressed by order of Cromwell). The condition the Scots were kept in were utterly appalling. Records indicate that the Scots died at an average of 30 a day between 11 September and 31 October and it seems this reached over 100 a day with virtually no food, clean water or heat and the linked spread of disease and infection. In total a further 1,700 prisoners died. Their bodies were thrown into pits dug by the prisoners in and around Durham Cathedral and castle.
Of the 5,000 men who started the march from Dunbar only 1300 were still alive less than two months later. Nine hundred were sold as slaves and transported to the New World, (mainly Virginia, Massachusetts and the Barbados colony in the Caribbean.) 500 elected to fight with the French army against the Spanish. They were still fighting seven years later.
There is no memorial to these unknown Scottish soldiers (who died horrible deaths at the hands of their English captors in Durham Cathedral or Castle.) They lie in anonymity and without Christian burial (on foreign soil in the place they were imprisoned,) far from their homes and the graves of their loved ones.
Remains of Scottish soldiers uncovered at Durham Cathedral
https://wildernessofpeace.wordpress.com/2015/06/05/uppity-jocks-cant-you-take-a-joke/
http://www.dailymail.co.uk/sciencetech/article-3219504/Skeletons-mass-grave-belong-Scottish-prisoners-war-Durham-remains-belong-17th-century-defeated-Royalists.html
http://www.bbc.co.uk/news/uk-england-tyne-34116842
Skeletons of Scottish prisoners of war, from the Battle of Dunbar and subsequently murdered by Cromwell's soldiers have been discovered in a mass grave close to Durham cathedral
Author caltonjockPosted on November 25, 2016 Categories Uncategorized
One thought on "Lucy Frazer – Tory MP – Destined for Greatness – Jewess – Strangely Seemingly an Admirer of Perpetrators of Shoahs – But Only if Scots Are on the Receiving End of the Brutality"
Frank Hocking says:
The Germans obviously missed one. should have been 6 million and one. The bitch.
Don't see why a Jewish woman should be in the UK government anyway. because they are worse than Muslims .but then you put trash with trash you don't mix it with good stuff. you can tell what she come from. by the nose.is like a battering ram Cromwell used. pity the humor she obviously delights in knowing that Cromwell's army slaughtered thousands of Scots at Dunbar. And she delights in making comments as long as it is Scots. then she does not mind. pity it wasn't Jews. Then we would see the other side of her pathetic face. When you see it trying to smile .its more like a Tsunami. she needs to get something done with that mouth, is fucking massif. She is a down right idiot. she laughs at her own pathetic jokes. to the delight of the Tory porkers. What a disgusting pathetic group of mixed race in any one Government. No wonder we want to see the end of them. Let Guy Fox finish what he started.
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Calomys callosus är en däggdjursart som först beskrevs av Rengger 1830.  Calomys callosus ingår i släktet aftonmöss och familjen hamsterartade gnagare. IUCN kategoriserar arten globalt som livskraftig. Inga underarter finns listade.

Denna gnagare förekommer i östra Bolivia, centrala till södra Brasilien, Paraguay och norra Argentina. I bergstrakter når arten 2050 meter över havet. Habitatet utgörs av torra eller halvtorra buskskogar, savanner med glest fördelade träd och skogsgläntor.

Honor föder i genomsnitt 4,5 ungar per kull efter cirka 22 dagar dräktighet. Ungarna föds blinda och de väger bara något över 2 gram vid födelsen. Två veckor efter födelsen eller något senare slutar honan med digivning.

Källor

Aftonmöss
callosus
Däggdjur i neotropiska regionen

redpajama_set_name: RedPajamaWikipedia


PORT WASHINGTON, N.Y. (BRAIN) — The NPD Group has expanded its Retail Tracking Service to provide a more comprehensive view of the U.S. cycling market, adding coverage of all retail channels for bicycle products, including outdoor specialty, sporting goods, national chains, mass merchants, and e-commerce.
MOUNT PROSPECT, Ill. (BRAIN) — The National Sporting Goods Association found that consumer-reported sales of sporting goods equipment, athletic footwear and athletic apparel was up 4 percent in 2015. The NSGA published its 2016 edition of its Sporting Goods Market report this week.
BOULDER, Colo. (BRAIN) — Supplier sales to U.S. retailers from January through March showed no improvement over the same period a year earlier, in part contributing to a 40 percent jump in supplier inventory levels at the end of March, according to the Bicycle Product Suppliers Association.
ANAHEIM, Calif. (BRAIN) — Triathlon Business International is conducting its second annual Industry Confidence Survey, which the trade group said will serve as "a barometer to multisport industry sentiment on key issues ranging from perspectives on retail prices to the general outlook for growth of triathlon in the U.S."
BOULDER, Colo. (BRAIN) — The NPD Group says IBD bike sales were up 4 percent in dollars and 7 percent in units in the first nine months of 2015.
"Contrary to the first nine months of 2014, the cycling industry is on the plus side heading into this holiday season," the retail tracking company said.
WASHINGTON (BRAIN) — Sales of snow sports products were up 3 percent in August and September compared to last year, and the industry's trade association said the uptick was largely due to pent up demand in the West, where ski areas experienced a snow drought the last several seasons.
Gluskin Townley report finds shop visits have declined 17 percent since 2012.
ALLENTOWN, Pa. (BRAIN) — Bike shop visits by consumers declined 17 percent between 2012 and 2014, according to a new report based on consumer research by the industry consulting group Gluskin Townley.
Store pickup of online orders is increasingly popular, retail group says.
WASHINGTON (BRAIN) — The National Retail Federation is predicting that consumer holiday spending will be up a bit this year, to $805 per person, the highest amount in the 14 years that the NRF has been conducting its Holiday Consumer Spending Survey.
PORT WASHINGTON, N.Y. (BRAIN) — The NPD Group says IBD bicycle dollar sales grew 6 percent in the 12 months ended in August, driven in large part by growth in sales of electric, hybrid and mountain bicycles.
BOULDER, Colo. (BRAIN) — While women's specific bike, accessory and apparel sales are trending upward, there is still a big opportunity to grow.
TBI study includes spending on travel, accommodations and other expenses.
ORANGE, Calif. (BRAIN) — Triathlon Business International has completed an extensive study of the consumer triathlon market and its economic impacts.
MOUNT PROSPECT, Ill. (BRAIN) — The National Sporting Goods Association found that consumer-reported retail sales of sporting goods equipment, athletic footwear and athletic apparel increased 2 percent in 2014. The NSGA published its 2015 edition of its Sporting Goods Market report this week.
BOULDER, Colo. (BRAIN) — PeopleForBikes says new analysis of its research into American bicycling shows that many assumptions about women and bikes are wrong.
MOUNT PROSPECT, Ill. (BRAIN) — The number of Americans who rode six or more days remained flat last year compared to 2013 figures, according to the National Sporting Goods Association, which tracks participation trends for cycling and 53 other sports and recreational activities.
SILVER SPRING, Md. (BRAIN) — The Sports & Fitness Industry Association published the first 25 of its 94 single sport and fitness participation reports Monday. These reports provide an analysis and breakdown of participation demographics by specific sport or fitness activity, including bicycling, triathlon and stationary cycling.
Outdoor Foundation attributes 0.8% decline to extreme weather but says nearly half of Americans still got outside in 2014.
WASHINGTON (BRAIN) — Nearly half of all Americans — 48.4 percent — participated in at least one outdoor activity in 2014, a decrease of 0.8 percent compared with 2013, according to the Outdoor Foundation's 2015 Outdoor Recreation Participation Topline Report.
NEW YORK (BRAIN) — National real estate brokerage firm Redfin unveiled its ranking of bikeable cities Thursday to coincide with Bike to Work week.
SILVER SPRING, Md. (BRAIN) – The Sports & Fitness Industry Association this week released its 2015 participation report that ranked sports activities by number of participants. Road bicycling ranked 7th on the survey's top 10 list with 20.4 million core participants in 2014.
Fat bikes and 650b front-suspension bike sales were up, as were sales of sub-$1,500 road bikes.
BOULDER, Colo. (BRAIN) — U.S. bike wholesaler sales grew 6 percent through March compared with the first three months of last year, according to the Bicycle Product Suppliers Association wholesale business report released this week. Sales were up more than 7 percent in March.


text:
Roland Barthes' Tokyo: "Empire of Signs" Fifty Years Later
Roland Barthes first visited Japan in 1966, not long after the defeated and reconstructed country announced its return to the international community with the 1964 Summer Olympics in Tokyo. Japan would have hosted its first Olympic Games there in 1940, had World War II not caused the duty to pass to Helsinki. Now, half a century after the semiotician from Cherbourg walked its streets, the Japanese capital prepares for its second Summer Olympics in 2020, in hopes of signaling another reemergence: not from wartime devastation by the most advanced weapons known to man, but from the long hangover of the postwar economic bubble, burst in the 1990s, and the subsequent "lost decade" now turning into a lost quarter century.
By most economic and demographic indicators, Japan has long looked like a country in trouble, even though a foreign visitor sees signs of robust health everywhere: a refined and efficient service culture; reliable infrastructure; conspicuous displays of high technology; shops filled with an astonishing amount and variety of carefully designed products; lively packs of uniformed schoolchildren, the smallest of whom ride on the back seats of their mothers' bicycles. The aftermath of 2011's Tōhoku earthquake may have exposed deep and previously unsuspected societal frailties, and yet, on every one of my trips to Japan I marvel at all those well-put-together moms calmly biking their kids to school. Surely they indicate an achievement of which the rest of the developed world, no matter its wealth, can only dream — even if on paper the country itself looks about to lie down and die.
As a pioneer in the study of signs and symbols, Barthes would have enjoyed grappling with all the conflicting signals sent out by 21st-century Japan. He lived through most of the postwar years when the Japanese economy grew at an unprecedented rate, but he missed the downright grotesque inflation of Japanese asset prices in the decade after his death in 1980. By then the West, and especially the United States, nervously fixated on images of flush Japanese tourists landing in Hawaii and buying mansions in cash, sharp-suited Japanese businessmen lavishly entertaining on sinisterly vast expense accounts, and Croesan Japanese conglomerates snapping up Los Angeles's movie studios and downtown high-rises.
Read the whole thing at the Los Angeles Review of Books.
This was written by Colin Marshall. Posted on Saturday, December 31, 2016, at 7:03 am. Filed under books, Japan, Los Angeles Review of Books, Tokyo. Bookmark the permalink. Follow comments here with the RSS feed.	Comments are closed, but you can leave a trackback.
‹ I talk about "Who Framed Roger Rabbit?" and the Los Angeles streetcar conspiracy on KPCC
My ten favorite Open Culture posts of 2016: Steely Dan's Recording Techniques, Nam June Paik's New Year Special, the Philosophy of Bill Murray, and More ›

redpajama_set_name: RedPajamaCommonCrawl


* Up to 5 display images from FURUNO Radar FAR-3000/FCR-2xx9 series or FURUNO ECDIS FMD-3200/3300 can be stored simultaneously through Ethernet without optional SSD.
* Radar and ECDIS images can be recorded at 15-second intervals.
* Radar and ECDIS that utilizes LAN interface based on IEC 61162-450 can be connected through Ethernet.
* Necessary when FURUNO Radar FAR-15x8/FAR- 2xx7, ECDIS FEA-2107/2807 or the products of third-party manufacturers are connected to VR-7000/7000S.
Multiple sensor adapters can be interfaced depending upon the number of sensors to be interfaced.
If marine incidents occur, record the data, which are recorded to the capsule, to the safe storage area for a long term recording by this function.
The onboard sensors can be integrated and interfaced to the VR-7000/7000S thanks to Video LAN converter and sensor adapters.
Also, the collected data can be replayed with the PC software or extracted onto a USB flash memories for later analysis.
This software extracts and displays the data accumulated from the VR-7000/7000S, in real time, on the networked PC screen. Also, the data can be replayed for a more thorough data analysis at a later date.
At-a-glance recognition of VDR status with minimal operation.


Looks like we've made it through another year. Launching this blog is my celebration.
Every menu item has at least one post so far, except Viewpoints. I plan to use Viewpoints to comment occasionally on events in the news and culture. If you have a suggestion for a topic for the inaugural post under Viewpoints, feel free to let me know, either in a Comment or by emailing me at kathh2015@gmail.com.
I wish for everyone a healthy and fulfilling new year.


text:
Phone42.com / Smartphones / ZTE / ZTE Blade Q Plus
ZTE Blade Q Plus (2015) review
Android OS v4.4.4 (KitKat)
ZTE announced the launch of the Blade Q Plus smartphone in 2015. The manufacturer is offering the ZTE Blade Q Plus phone with 4.5 inches display, Qualcomm Snapdragon 410 MSM8916 chipset, 1 GB of RAM and 8 GB of internal storage.
The ZTE Blade Q Plus comes with Android OS v4.4.4 (KitKat) out of the box, and a Li-Ion 2000 mAh battery. The 4.5 inches phone comes with TFT display (480 x 854 px). A thin-film transistor (TFT) is a type of field-effect transistor that is usually used in a liquid crystal display (LCD). This type of display features a thin-film transistor for each individual pixel.
Add ZTE Blade Q Plus to your comparison
ZTE Blade Q Plus specifications
Blade Q Plus
BQPLUS
The weight of ZTE Blade Q Plus is about 4.9 oz with battery, which is average compared to most of the smartphones with same size. The dimensions of this ZTE phone are average (5.2 x 2.64 x 0.39 inches).
The ZTE Blade Q Plus comes with Android OS v4.4.4 (KitKat) out of the box, but it is upgradable to the next OS.
The size of Blade Q Plus screen is measured diagonally, from corner to corner. The 4.5 inches TFT display of ZTE Blade Q Plus with 480 x 854 px resolution is multitouch capable.
The internal memory of ZTE Blade Q Plus (1 GB) cannot be expanded, but the internal storage (8 GB) can be expanded with a compatible microSD card.
The camera of the Blade Q Plus is equipped with autofocus. An autofocus (or AF) optical system uses a sensor, a control system and a motor to focus on an automatically or manually selected point or area. Very few Android cameras come without autofocus.
The ZTE Blade Q Plus supports 2G, 3G and 4G/LTE networks. This phone has a built-in GPS. GPS is a highly accurate navigation system using signals from satellites to determine a location on the Earth's surface, irrespective of weather conditions. Most phones have a GPS unit on board. The Blade Q Plus phone has built-in FM radio. The built-in FM Radio is great for listening to your favorite radio station.
The Li-Ion 2000 mAh battery gives the smartphone a good battery backup. The Li-ion is a low maintenance battery which offers high energy, is lightweight and does not require periodic full discharge. The battery of Blade Q Plus is removable. You can simply buy a new piece from the open market and interchange it easily by yourself.
The specifications above are based on the official data published by the manufacturer, but we also take user reviews into consideration. If you found an error or something lacking in the specifications above for the ZTE Blade Q Plus, then don't hesitate and signal the problem to us.
ZTE Blade Q Plus pictures


text:
Q: Getting the HTML Markup from a dynamically constructed div How can I get the markup for something I created like this
Div MyDiv = new Div("Test");

I essentially want to get a string with the contents
string SomeString = "
Test
";

Or maybe even
string SomeString = "
Test
;

Is there any way I can do that?

A: One option is to give the div an id and a runat="server" attribute.  If\when the page is posted back to the codebehind, use the InnerHTML property.  Example:

And in your codebehind
string contents = myDiv.InnerHTML;

If you only need this data client side, use javascript, like mentioned in the comments above.  You can use the following code:
var x = document.getElementById('myDiv').innerHTML;

If you already have jQuery, then:
var x = $('myDiv').html();

redpajama_set_name: RedPajamaStackExchange


Kerry grew up outside of NYC before braving the cold winter weather at the University of New Hampshire. She is a Certified Athletic Trainer and specializes in injury prevention and management, as well as rehabilitative exercise. Eventually she wised up and traded blizzards for the beach; which landed her in Charleston! She is a dog enthusiast who does not yet have a little pup of her own and when she is not at Fleet Feet you can find her at the ACS Hope Lodge downtown! At Fleet Feet Kerry is coordinating Monday Rundays as well as Medical Outreach in the Community. She is also co-coordinating our Mt. Pleasant Pub Runs.


If the goal is to get hired as a designer, then you have to mingle and circulate on social media while constantly learning and working on your designs. Although it's necessary to have the gift and ability to create original work, it's all insufficient without software related knowledge and experience.... He wants to see more than "just an Illustrator vector whacked on a LiveImage Photoshop file. Show me how the work goes deep." Show me how the work goes deep." And just for clarification, yes, you should always bring your resume and portfolio with you to the interview, even if you've submitted them before.
However, with formal education and a professional portfolio, it can be easier to get jobs that use a talent for drawing, such as animator or illustrator. Learning useful software, such as photo editing and animation programs, is another way to turn drawing ability into a career.
Hi, I'm Aiah, an illustrator, Art director, Designer living in Alexandria, Egypt. I am 1. an illustrator with 2. a penchant for Animation and a bit design that acutely 3.
This year is all about innovations and originality, hence, don't be afraid to experiment with colors, design, and software. Of course, programs such as Photoshop, Illustrator, CorelDraw, and GIMP will remain the basic essentials for both novice and graphic design experts.
If the goal is to get hired as a designer, then you have to mingle and circulate on social media while constantly learning and working on your designs. Although it's necessary to have the gift and ability to create original work, it's all insufficient without software related knowledge and experience.


text:
package ikube.action.index.analyzer;

import ikube.IConstants;
import org.apache.lucene.analysis.Analyzer;
import org.apache.lucene.analysis.Tokenizer;
import org.apache.lucene.analysis.core.LowerCaseFilter;
import org.apache.lucene.analysis.ngram.NGramTokenFilter;
import org.apache.lucene.analysis.ngram.NGramTokenizer;

import java.io.IOException;
import java.io.Reader;

/**
 * This analyzer will break the data into n-grams. This means that there will potentially be several times the number
 * of words and indeed in the data. This
 * facilitates fuzzy searching for sub-strings rather than wild cards which are very expensive,
 * but will add considerably to the size of the index.
 *
 * @author Michael Couck
 * @version 01.00
 * @since 29.10.12
 */
public final class NgramAnalyzer extends Analyzer {

	private int minGram = 4;
	private int maxGram = 21;

	protected TokenStreamComponents createComponents(String fieldName, Reader reader) {
		// Tokenizer tokenizer = new LowerCaseTokenizer(IConstants.LUCENE_VERSION, reader);
		Tokenizer tokenizer = new NGramTokenizer(IConstants.LUCENE_VERSION, reader, minGram, maxGram);
		NGramTokenFilter nGramTokenFilter = new NGramTokenFilter(IConstants.LUCENE_VERSION, tokenizer, minGram,
			maxGram);
		LowerCaseFilter lowerCaseFilter = new LowerCaseFilter(IConstants.LUCENE_VERSION, nGramTokenFilter);
		return new TokenStreamComponents(tokenizer, lowerCaseFilter) {
			@Override
			protected void setReader(final Reader reader) throws IOException {
				super.setReader(reader);
			}
		};
	}

	public void setMinGram(int minGram) {
		this.minGram = minGram;
	}

	public void setMaxGram(int maxGram) {
		this.maxGram = maxGram;
	}

}

redpajama_set_name: RedPajamaGithub


Posted on Oct 30, 2015 by Appleton Downtown Inc.
The Exhibition Center project took a very important step last night!
I have again attached those documents for your convenience.
Last night was a very important step in this process.
Mayor Hanna strongly believed, and other municipalities concurred, that the Appleton Common Council needed to take the leadership vote on the IGA documents prior to the other municipalities posting these for action on their agenda. The vote last night will allow the other nine municipalities to now discuss and vote on the IGA documents in the next couple of weeks.
The next steps for the Appleton Common Council are to review the Property Improvement Plan (PIP) for the Radisson Paper Valley and to discuss and vote on the Management Agreement (MA) between the City of Appleton and Inner Circle.
Once the Appleton Common Council Agenda is posted with the PIP and MA I will share those documents via this communication method.
Thank you to everyone who has taken your time to review documents, ask questions, share your concerns and thoughts. Collaboration makes the process and the project stronger!


text:
Gabi foi um programa de entrevistas apresentado por Marília Gabriela e produzido pela RedeTV!, e exibido de 5 de junho de 2000 a 30 de abril de 2002.

Após sair do SBT, Marília Gabriela conversou com diversas emissoras como a Rede Bandeirantes, mas acabou acertando com a RedeTV! na metade de abril de 2000. "Eles me queriam lá para fazer um programa noturno de entrevistas. [...] Expus, então, a idéia [sic] de casar tevê e Internet e eles toparam na hora".

De terça a sexta, o programa era exibido às onze horas da noite e, às segundas — por causa do humorístico Te Vi na TV, de João Kléber, que entrava no ar nesse mesmo dia, às onze, Gabi era transmitido à meia noite. Às terças e quartas-feiras, o programa era ao vivo, enquanto nas segundas-feiras, por entrar mais tarde, tratava-se sobre sexo. Inicialmente o programa se chamaria Gabi.com, mas foi renomeado para apenas Gabi pois a apresentadora pensou nas pessoas que não tinham internet em casa. Os telespectadores podiam fazer perguntas por e-mail, telefone ou fax. Além disso, Marília Gabriela se relacionava com a audiência também por meio de fotos. Com uma câmera digital, ela conseguia imagens peculiares de seus entrevistados, a pedido dos telespectadores.

Gabi
Programas de entrevistas do Brasil
Programas de televisão do Brasil que estrearam em 2000
Programas de televisão do Brasil encerrados em 2002
Programas de televisão em língua portuguesa


PCT is a world-class training facility that specializes in teaching young women and men life lessons through the sport of cheerleading. Life lessons such as: how to be confident and capable, work ethic, sportsmanship, community service, teamwork, trust, listening skills, time management, goal setting, winning and losing with grace, and so much more. Our families are raving fans about our program because of the benefits they see week-in and week out at PCT.
We've got an All-Star team for everybody! New to cheer, new to PCT or a 10-year veteran, if YOU have the desire, and the work ethic then we've got the PERFECT team for you.
People choose PCT for a variety of reasons: our incredible facility that allows us to comfortably have multiple teams training. Or knowing that your child is being put on a path to reach their fullest potential – we've got the perfect team for your child this season AND for next season. With teams at every single level and age division, you know your child will be perfectly placed with an eye to their future development.
You've got big dreams for your child? We do too. Seeing every athlete progress under the tutelage of our amazing staff and world-class training program is the reason we exist!
So how do you become part of the PCT All Star Program?
First things first. Get on our mailing list! You'll get updates about FREE clinics, Open Gyms and helpful information about Team Placements. Fill out the form below and you'll be redirected to our Information page.
All Star Novice is THE PERFECT introduction into the exciting world of All Star Cheerleading!
Do you know someone who wants to be introduced to the sport of all star cheerleading? PCT's All Star Novice program is a great stepping stone to the sport of all star cheer and is perfect for any age or level. Our program provides athletes an opportunity to participate in competitive cheerleading at its best from the best, at an affordable price. Athletes will get the chance to compete three times locally and get to see what being part of the PCT Family is all about!


text:
Director of Alumni Relations
Director of Alumni Relations Salary in North Dakota
How much does a Director of Alumni Relations make in North Dakota? The average Director of Alumni Relations salary in North Dakota is $68,794 as of December 26, 2019, but the range typically falls between $60,236 and $78,040. Salary ranges can vary widely depending on the city and many other important factors, including education, certifications, additional skills, the number of years you have spent in your profession.
Salary range for an Director of Alumni Relations
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Location Mandan, ND Avg. Salary $68,288 Date Updated December 26, 2019
Location West Fargo, ND Avg. Salary $68,735 Date Updated December 26, 2019
Location Abercrombie, ND Avg. Salary $67,636 Date Updated December 26, 2019
Location Absaraka, ND Avg. Salary $67,636 Date Updated December 26, 2019
Location Adams, ND Avg. Salary $62,889 Date Updated December 26, 2019
Location Agate, ND Avg. Salary $65,039 Date Updated December 26, 2019
Location Alamo, ND Avg. Salary $67,189 Date Updated December 26, 2019
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Location Alsen, ND Avg. Salary $62,889 Date Updated December 26, 2019
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Elwood Staffing - Dickinson , ND
As directed by supervisor, performs tasks associated with Kitting, Kanban point-of-use ... Determine the quality and quantity of materials received in relation to prescribed specifications
Elwood Staffing - 10 days ago
What does a Director of Alumni Relations do?
Director of Alumni Relations in Fort Wayne, IN
Develop, manage and monitor the components of the Association budget that are related to marketing, membership recruitment and activities, and member benefits programming.
Director of Alumni Relations in Hanover, NH
Works with the Executive Director of Advancement Administration to identify and implement the systems and data needs required to support a successful program, including the gathering and maintenance of accurate alumni information and the implementation of technology to enhance the effectiveness of alumni programming.
Serves as editor of the Alumni E-Newsletter and, with the support of the Communications department, helps generate content, provides class notes, coordinates the production and meeting the monthly deadline for the distribution of the newsletter.
Demonstrated ability to manage projects, working successfully with clients and stakeholders to produce high-quality events that meet budget and time constraints, and keep colleagues, vendors, faculty, volunteers and other clients motivated and well served.
Provides, promotes and executes programs and events (local/regional/national) for a broad range of alumni constituencies which aim to increase overall alumni giving and involvement that supports the alumni strategic plan.
Plan, implement, and/or provide tools and resources for alumni events whether statewide or at the campus level, as well as meaningful volunteer and mentoring opportunities.
Assist area alumni in implementing regional gatherings while supporting and encouraging the development of local leadership within alumni chapters.
Collaborate with deans, academic chairs, and program directors, development staff, members of the Communications & Marketing team, and other departments with focus on delivering university news, executing alumni-centered programming and facilitating multi-faceted opportunities for alumni to give back.
Manages alumni organizations, including the Villanova Law Alumni Association (VLAA), Minority Alumni Society (MAS), and NovUs, advising and supporting the Officers and Board of Advisors in its role of developing alumni affinity in support of the advancement of the Law School and its community.
Create, implement, and communicate a comprehensive alumni engagement plan that strategically engages alumni and connects them to their alma mater, including events, programming, networking and volunteer opportunities including (but not limited to) Reunion and Volunteer Work Project.
North Dakota is a Midwestern state of the United States. It borders Canada and lies at the center of the North American continent. The geographic center of North America is near the town of Rugby. Bismarck is the capital of North Dakota, and Fargo is the largest city. Soil is North Dakota's most precious resource. It is the base of the state's great agricultural wealth. But North Dakota also has enormous mineral resources. These mineral resources include billions of tons of lignite coal. In addition, North Dakota has large oil reserves. Petroleum was discovered in the state in 1951 and quickly...
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About North Dakota North Dakota is a Midwestern state of the United States. It borders Canada and lies at the center of the North American continent. The geographic cent....More


text:
\section{Introduction}

\smallskip
Throughout the paper we assume the Riemann Hypothesis (RH) for the Riemann zeta function $\zeta(s)$. Let
\[
F(X,T) = 4 \sum_{-T\leq\gamma,\gamma'\leq T}
\frac{X^{i(\gamma-\gamma')}}{4+(\gamma-\gamma')^2},
\]
where $\gamma,\gamma'$ run over the imaginary part of the non-trivial zeros of $\zeta(s)$, be the pair-correlation function of the zeta zeros.  In his famous paper \cite{Mon/1973}, Montgomery conjectured that
\begin{equation}
\label{1-1}
F(X,T) \sim \frac{T}{\pi}\log T
\end{equation}
as $X\to\infty$, uniformly for $X^\varepsilon \leq T \leq X$ for any fixed $0<\varepsilon <1$. Moreover, the essentially equivalent assertion in \cite{Mon/1973}, that for any fixed $0<\alpha<\beta$ and $T\to\infty$
\[
\frac{\pi}{T\log T} \sum_{\substack{-T\leq\gamma,\gamma'\leq T\\ \frac{2\pi\alpha}{\log T} \leq \gamma-\gamma'\leq \frac{2\pi\beta}{\log T}}} 1 \sim \int_\alpha^\beta \Big(1-\big(\frac{\sin\pi u}{\pi u}\big)^2\Big) \mathrm{d} u,
\]
raised an interesting connection with the Random Matrix Theory; see the excellent survey by Conrey \cite{Con/2001}.  A sharp numerical confirmation of the latter conjecture was given by Odlyzko \cite{Odl/1987},\cite{Odl/1989}; see also \cite{Odl/web}.

\medskip
The pair-correlation conjecture \eqref{1-1}, as well as its variants, found several applications in prime number theory. Among the vast literature on this subject we recall here the papers by Gallagher $\&$ Mueller \cite{Ga-Mu/1978}, Heath-Brown \cite{HB/1982}, Goldston $\&$ Montgomery \cite{Go-Mo/1987} and Rudnick $\&$ Sarnak \cite{Ru-Sa/1996}. In particular, writing
\[
J(X,h) = \int_X^{2X} (\psi(x+h)-\psi(x)-h)^2 \mathrm{d} x \quad \text{and} \quad \psi(x) = x + \Delta(x),
\]
Heath-Brown \cite{HB/1982} showed that assuming RH and various forms of the pair-correlation conjecture, weaker than \eqref{1-1} concerning the size and/or the uniformity, one could get sharp bounds for the error term $\Delta(x)$, for the mean-square of primes in short intervals $J(X,h)$ and for the difference between consecutive primes $p_{n+1}-p_n$. The ideas and techniques in \cite{HB/1982} were further developed by several authors. We mention here the papers by Heath-Brown $\&$ Goldston \cite{HB-Go/1984} and Liu $\&$ Ye \cite{Li-Ye/2001}, the latter assuming RH and a version of the pair-correlation conjecture given by Rudnick $\&$ Sarnak \cite{Ru-Sa/1996}. We also mention the earlier paper by Mueller \cite{Mue/1981} on $p_{n+1}-p_n$. Moreover, we recall that Gonek \cite{Gon/1993} proposed a conjecture related to Landau's explicit formula and showed that such a conjecture has very strong consequences on the distribution of primes in short intervals. We shall enter the contents of the above quoted papers later on, while comparing to our results.

\medskip
It turns out that the applications of the pair-correlation conjecture contained in some of the above papers, along with further results, can be framed in a unified way as consequences of suitable assumptions on the extended pair-correlation function
\[
F(X,T,\tau) = 4 \sum_{-T\leq\gamma,\gamma'\leq T} \frac{X^{i(\gamma-\gamma')}}{4+\tau^{2}(\gamma-\gamma')^2},
\]
where $0\leq \tau \leq 1$. Note that at the extreme $\tau$-values we have $F(X,T,1) = F(X,T)$ and $F(X,T,0) = |\Sigma(X,T)|^2$, where
\begin{equation}
\label{1-2}
\Sigma(X,T) = \sum_{|\gamma|\leq T} X^{i\gamma}
\end{equation}
is the exponential sum over zeros appearing in Landau's explicit formula, see Gonek \cite{Gon/1993}. We first give an interpretation of $F(X,T,\tau)$ as a suitable pair-correlation function. Indeed, it is clear that the numbers $\widetilde{\rho}=\tau/2 + i\tau\gamma$ are the non-trivial zeros of the function $Z(s) = \zeta(s/\tau)$, and
\begin{equation}
\label{1-3}
F(X,T,\tau) = 4 \sum_{-\tau T\leq \tau\gamma,\tau\gamma'\leq \tau T} \frac{(X^{1/\tau})^{i(\tau\gamma-\tau\gamma')}}{4+(\tau\gamma-\tau\gamma')^2}.
\end{equation}
Moreover, given a rather general $L$-function $L(s)$ (namely, belonging to the Selberg class of $L$-functions, see e.g. Kaczorowski $\&$ Perelli \cite{Ka-Pe/1999b}) and assuming the Riemann Hypothesis for $L(s)$, Murty $\&$ Perelli \cite{Mu-Pe/1999} investigated the pair-correlation function $F_L(X,T)$ which, with a slightly different normalization with respect to \cite{Mu-Pe/1999}, is defined in analogy with $F(X,T)$ as
\begin{equation}
\label{1-4}
F_L(X,T) = 4 \sum_{-T\leq\widetilde{\gamma},\widetilde{\gamma}'\leq T}
\frac{(X^d)^{i(\widetilde{\gamma}-\widetilde{\gamma}')}}{4+(\widetilde{\gamma}-\widetilde{\gamma}')^2}.
\end{equation}
Here the non-trivial zeros of $L(s)$ are of the form $\widetilde{\rho} = 1/2 + i\widetilde{\gamma}$, and $d$ is the degree of $L(s)$ (i.e. twice the sum of the coefficients $\lambda_j$ in the $\Gamma$-factors $\Gamma(\lambda_js+\mu_j)$ appearing in its functional equation; see \cite{Ka-Pe/1999b}). Since $\zeta(s)$ has degree 1, we have $F_\zeta(X,T)=F(X,T)$. The functional equation of $Z(s)$, obtained at once from the one of $\zeta(s)$, suggests that we may heuristically regard $Z(s)$ as an $L$-function of degree $d$ and conductor $q$ given by
\begin{equation}
\label{1-5}
d=\frac{1}{\tau} \hskip1.5cm q=\big(\frac{1}{\tau}\big)^{\frac{1}{\tau}}
\end{equation}
(see again \cite{Ka-Pe/1999b} for the definition of conductor), although $Z(s)$ does not belong to the Selberg class if $\tau\neq 1$ (e.g. since it has a pole at $s=\tau$). Hence in view of \eqref{1-3}, \eqref{1-4} and \eqref{1-5} we have
\begin{equation}
\label{1-6}
F(X,T,\tau) = F_Z(X,\tau T).
\end{equation}
The pair-correlation conjecture in Murty $\&$ Perelli \cite{Mu-Pe/1999}, namely
\[
F_L(X,T) \sim \frac{dT}{\pi} \log T
\]
as $X\to\infty$, uniformly for $X^\varepsilon \leq T \leq X$ for any fixed $0<\varepsilon<1$, is meant for a given $L$-function, hence in particular for fixed degree $d$. In view of the Riemann-von Mangoldt formula for the the Selberg class, a reasonable $(d,q)$-uniform version of this conjecture is
\begin{equation}
\label{1-7}
F_L(X,T) \sim \frac{dT}{\pi} \log q^{1/d}T
\end{equation}
in the same range of uniformity for $X$ and $T$. Hence, in view of \eqref{1-5}, \eqref{1-6} and \eqref{1-7} we may expect that
\begin{equation}
\label{1-8}
F(X,T,\tau) \ll T\log T
\end{equation}
uniformly for $X^\varepsilon \leq \tau T \leq X$, and possibly even in wider ranges for $T$ and $\tau$.  We refer to \eqref{1-7} as the general Montgomery conjecture.

\medskip
We recall that the asymptotic behavior of $F(X,T)$ in \eqref{1-1} is expected to change around $T=X$ and that, under RH, Montgomery \cite{Mon/1973} (and Goldston \cite{Gol/1981}) detected the behavior of $F(X,T)$ when $T\geq X$. An analog of the latter result is proved in Murty $\&$ Perelli \cite{Mu-Pe/1999} for the functions $F_L(X,T)$; in such case the assumption $T\geq X^d$ is required. However, in our case the function $Z(s)$ is a $\tau$-rescaling of $\zeta(s)$, and for this reason we may expect that the role of $X$ is played essentially by $X^\tau$. In the companion paper \cite{L-P-Z/tauPC-II} we obtain, assuming certain hypotheses, the asymptotic behavior of $F(X,T,\tau)$ in suitable ranges of $T$ and $\tau$, when $T\geq X$. Note that the trivial bound for $F(X,T)$ is $O(T\log^2T)$ uniformly for $X,T\geq 2$, hence only $\log T$ worse than the expected size (when $X^\varepsilon\leq T\leq X$), while the trivial bound for $F(X,T,\tau)$ is for small $\tau$ much worse than the expected size (again if $T\leq X$). Indeed, we have
\begin{equation}
\label{1-9}
F(X,T,\tau) \ll \min \Bigl(T; \frac{1}{\tau}\Bigr)\ T \log^2 T
\end{equation}
uniformly for $X,T\geq 2$ and $0\leq \tau\leq 1$. This is obvious for $\tau\leq 1/T$, while in the opposite case we have
\[
F(X,T,\tau) \ll T\log^2T \sum_{m=0}^{T}  \frac{1}{1+\tau^2m^2} \ll \frac{1}{\tau} T\log^2T.
\]
In view of \eqref{1-8} and \eqref{1-9}, in this paper we also consider the slightly more conservative pair-correlation conjecture
\begin{equation}
\label{1-10}
F(X,T,\tau) \ll TX^\varepsilon
\end{equation}
for every $\varepsilon>0$, with $T$ and $\tau$ in suitable ranges.

\medskip
The extended pair-correlation function $F(X,T,\tau)$ already appears in Heath-Brown $\&$ Goldston \cite{HB-Go/1984} (as $G_\beta(X,T)$, $\beta=1/\tau$, in their notation) and is used there to study the size of $p_{n+1}-p_n$ under Montgomery's conjecture \eqref{1-1} with $T$ in a certain range. To this end, $F(X,T,\tau)$ is related to $F(X,T)$ (see Lemma 2 in \cite{HB-Go/1984}), and the required bounds on $F(X,T,\tau)$ are obtained from the asymptotic formula for $F(X,T)$. However, the information obtained on $F(X,T,\tau)$ in that way reflects its expected order of magnitude only in a rather short range of $\tau$ close to 1. Clearly, that approach allows to obtain bounds for $F(X,T,\tau)$ from asymptotic formulae with remainder for $F(X,T)$, say of the form considered in our recent paper \cite{L-P-Z/2012} (roughly, an error of type $T\varepsilon(T)$ in input gives an error of type $T\varepsilon(T)/\tau^2$ in output). We remark that, in \cite{HB-Go/1984} and in subsequent works on the subject, the function $F(X,T,\tau)$ is used essentially as an intermediate tool in order to get information on the distribution of primes from suitable forms of Montgomery's pair-correlation conjecture. Our aim in this paper is, instead, to show how $F(X,T,\tau)$ directly controls the distribution of primes, especially in short intervals, depending on the range of uniformity in $T$ and $\tau$ allowed  in \eqref{1-10} or \eqref{1-8}. Actually, we mainly focus on the distribution of primes in short intervals of length $h\geq X^\varepsilon$, with $\varepsilon>0$ arbitrarily small, under the following hypotheses on the uniformity ranges for $T$ and $\tau$ where \eqref{1-10} is assumed to hold.

\medskip
{\bf Hypothesis $H(\eta)$.} {\sl Assume RH, let $0\leq \eta<1$ be fixed and let $X\to\infty$. If $0<\eta<1$ then \eqref{1-10} holds uniformly for $X^\eta \leq T \leq X$ \and $X^\eta/T \leq \tau \leq 1$; if $\eta=0$ then for every small $\varepsilon>0$, \eqref{1-10} holds uniformly for $X^\varepsilon \leq T \leq X$ \and $0 \leq \tau \leq 1$.}

\medskip
{\bf Remark 1.} First we note that for any fixed $0< \eta<1$, the uniformity range of $\tau T$ in hypothesis $H(\eta)$ is contained in $[X^\varepsilon, X]$, which in view of \eqref{1-6}-\eqref{1-8} may be regarded as a plausible range of a $d$-uniform version of the general Montgomery conjecture. When $\eta=0$, $\tau T$ may become $\leq X^\varepsilon$. However, as we remarked earlier, Gonek \cite{Gon/1993} made a conjecture on the size of the exponential sum $\Sigma(X,T)$ in \eqref{1-2}, namely
\begin{equation}
\label{1-11}
\Sigma(X,T) \ll T X^{-1/2+\varepsilon} + T^{1/2}X^\varepsilon
\end{equation}
for $X,T\geq2$. When $T\leq X$ the second term dominates, and hence in view of \eqref{1-11} and $F(X,T,0)=|\Sigma(X,T)|^2$ we may expect that for $X^\varepsilon\leq T \leq X$ and $0\leq \tau\leq 1$
\[
F(X,T,\tau) \ll |\Sigma(X,T)|^2 \ll TX^\varepsilon.
\]
Actually, by Lemma 5 below we have that for $\tau T\leq X^\varepsilon$
\[
F(X,T,\tau) \ll T^{1+\varepsilon} + X^\varepsilon \max_{t\leq T}|\Sigma(X,t)|^2,
\]
thus \eqref{1-10} follows from Gonek's conjecture in this range. Summarizing, we may say that our hypothesis on the size of $F(X,T,\tau)$ is supported by the general Montgomery conjecture in the range $X^\varepsilon\leq \tau T\leq X$, and by Gonek's conjecture in the remaining range $\tau T\leq X^\varepsilon$. Actually, $H(0)$ is equivalent to Gonek's conjecture restricted to the range $X^\varepsilon\leq T\leq X$. {\hfill \qedsymbol}

\medskip
Assuming RH and the pair-correlation conjecture in the form $F(X,T)\ll T\log T$ uniformly for $X^\varepsilon\leq T \leq X$, Heath-Brown \cite{HB/1982} implicitly proved (see the proof of Theorem 2 in \cite{HB/1982}) that $J(X,h) \ll hX\log(2X/h)$ uniformly for $1 \leq h \leq X^{1-\varepsilon}$. Later, Goldston $\&$ Montgomery \cite{Go-Mo/1987} proved that the conjectural asymptotic behavior of $F(X,T)$ implies that of $J(X,h)$, and vice versa. Assuming only RH one has $J(X,h) \ll hX\log^2(2X/h)$ uniformly for $1\leq h \leq X$, see Selberg \cite{Sel/1942} and Saffari $\&$ Vaughan \cite{Sa-Va/1977}. Let
\[
J(X,Y,h) = \int_X^{X+Y} \bigl( \psi(x + h) - \psi(x) - h \bigr)^2  \mathrm{d} x.
\]
We expect that, as $X\to\infty$,
\begin{equation}
\label{1-12}
J(X,Y,h) \ll hYX^\varepsilon
 \end{equation}
uniformly in suitable ranges of $h$ and $Y$, for every $\varepsilon>0$. We start with a result of Heath-Brown's type for $J(X,Y,h)$, under the assumption of hypothesis $H(\eta)$; we deal in \cite{L-P-Z/tauPC-II} with the analog of Goldston $\&$ Montgomery \cite{Go-Mo/1987}, i.e. with the equivalence between the asymptotic behavior of $J(X,Y,h)$ and $F(X,T,\tau)$.

\medskip
{\bf Theorem 1.} {\sl Assume $H(\eta)$. If $0<\eta<1$ then \eqref{1-12} holds uniformly for $1 \leq h \leq X^{1-\eta}$ and $hX^\eta\leq Y\leq X$. If $\eta=0$ then \eqref{1-12} holds uniformly for $1 \leq h \leq X^{1-\varepsilon}$ and $0\leq Y \leq X$, for every small $\varepsilon>0$.}

\medskip
By similar arguments one can get variants of Theorem 1, for example where both the hypothesis and the result are localized, or replacing $X^\varepsilon$ (both in input and in output) by suitable powers of $\log X$. Moreover,  a standard consequence of Theorem 1 is that
\[
\psi(x+h)-\psi(x)=h+O(h^{1/2}x^\varepsilon)
\]
for almost all $x\in[X,X+Y]$, with $h$ and $Y$ as in Theorem 1.

\medskip
Hypothesis $H(\eta)$ and Theorem 1 have further consequences on the distribution of primes in short intervals. For example, using in addition the inertia property of the function $\psi(x)$, from Theorem 1 we can deduce results on $\psi(x+h)-\psi(x)$ valid for all large $x$. On the other hand, hypothesis $H(\eta)$ can be used in a direct way for the same purpose. It is interesting to note that both approaches have advantages over the other, and the output of the two approaches is given by

\medskip
{\bf Theorem 2.} {\sl Assume $H(\eta)$ and let $\varepsilon>0$ be arbitrarily small. If $0<\eta<1/2-5\varepsilon$ then
\[
\psi(x+h)-\psi(x) = h +
\begin{cases}
O(h^{2/3}x^{\eta/3+\varepsilon}) &  \text{for} \  x^{\eta+5\varepsilon}\leq h\leq x^{1/2} \\
O(h^{1/3} x^{1/6+\eta/3+\varepsilon}) & \text{for} \  x^{1/2}\leq h\leq x^{1-\eta}.
\end{cases}
\]
If $\eta=0$ then $\psi(x+h)-\psi(x)= h+O( h^{1/2} x^\varepsilon)$ for $x^{3\varepsilon} \leq h \leq x^{1 - \varepsilon}$.}

\medskip
{\bf Remark 2.} Let $0<\eta<1/2-5\varepsilon$. The first estimate is obtained from Theorem 1 by means of the inertia property, while the second follows by a direct application of $H(\eta)$. Both estimates are non-trivial in the stated ranges for $h$, in the sense that the error terms are $O(\min(h^{1-\varepsilon},x^{1/2+\varepsilon}))$ and the ranges are non-empty (recall that the second bound holds under RH). However, we believe that the direct approach should always give a better result, namely
\[
\psi(x+h)-\psi(x)=h+O(h^{1/2}x^{\eta/2+\varepsilon})
\]
for $x^{\eta+4\varepsilon}\leq h\leq x^{1-\eta}$ and $0<\eta<1/2-2\varepsilon$. At present we cannot get such a bound because of the maximum over $t$ in Lemma 6 below, which forces the use of trivial bounds in a certain range, thus giving the weaker error term in Theorem 2. Note that there is a discontinuity between the results in the cases $\eta>0$ and $\eta=0$, due to an analogous discontinuity in hypothesis $H(\eta)$. Note also that hypothesis $H(\eta)$ with $\eta>0$ has two distinct consequences on primes in short intervals, one in mean-square (Theorem 1) and the other for each large $x$ (Theorem 2). The situation changes when $\eta=0$, since it is easily seen that in this case Theorem 1 is equivalent to Theorem 2. This is in agreement with the observation about Theorem 4 of Gonek \cite{Gon/1993} at the end of Remark 1, and indeed our bound for $\psi(x+h)-\psi(x)-h$ under $H(0)$ is the same as Gonek's bound under his conjecture. Finally, in the direction of Mueller \cite{Mue/1981}, Heath-Brown \cite{HB/1982}, Heath-Brown $\&$ Goldston \cite{HB-Go/1984} and Liu $\&$ Ye \cite{Li-Ye/2001}, bounds for $p_{n+1}-p_n$ of type $p_n^{1/2}f(p_n)$, with suitable $f(x)$'s, can be obtained from the estimate $F(X,T,\tau)\ll T\log T$ with suitable uniformity ranges of $T$ and $\tau$. In this case $T$ will be around $X^{1/2}$ and $\tau$ around a negative power of $\log X$. {\hfill \qedsymbol}

\bigskip
Now we turn to the problem of estimating $\Delta(x) = \psi(x)-x$ under suitable pair-correlation hypotheses. The bound $\Delta(x) = o(x^{1/2}\log^2x)$ has been deduced by Heath-Brown \cite{HB/1982} from the hypothesis $F(x,T)=o(T\log^2T)$ uniformly for $x^\varepsilon\leq T \leq x$, and the sharper estimate $\Delta(x) = O(x^{1/2}\log^{5/4}x)$ is stated in Liu $\&$ Ye \cite{Li-Ye/2001} as a consequence of a quantitative version of the general pair-correlation conjecture by Rudnick $\&$ Sarnak \cite{Ru-Sa/1996}, specialized to the case of the Riemann zeta function. However, \cite{Li-Ye/2001} contains several inaccuracies (in part detected by Goldston \cite{Gol/2002} and Chan \cite{Cha/2003}), hence the results are not reliable although the general strategy is clear. From the point of view of the function $F(X,T,\tau)$, the approach in \cite{HB/1982} and \cite{Li-Ye/2001} may be formalized as follows.

\medskip
{\bf Theorem 3.} {\sl Let $x\to\infty$, $Z=x^{1/2}\log^2x$ and $U=U(x)\in[10,Z]$. Assume RH and suppose that $F(x,T,\tau)\ll T\log T$ for some $\tau=\tau(x)\in[1/Z,1]$, uniformly for $U\leq T \leq Z$. Then}
\[
\Delta(x) \ll  x^{1/2}(\log^2U + \tau^{1/2} \log^{3/2}x).
\]

\medskip
We omit the proof of Theorem 3 as it follows along the lines of Theorem 1 of Heath-Brown \cite{HB/1982} and Theorem 2 of Liu $\&$ Ye \cite{Li-Ye/2001}, using Lemma 6 below and partial summation.

\medskip
{\bf Remarks.} {\bf 3.} The same argument allows a more general result, where the hypothetical bound on $F(x,T,\tau)$ is of type $O(T\log^a T)$ with $1\leq a \leq 2$, with suitable uniformity ranges. This produces a continuous chain of bounds for $\Delta(x)$ (exponents are linear in $a$), interpolating the classical result under RH and Theorem 3, in particular giving back Heath-Brown's bound.

{\bf 4.} Assume that the hypothesis in Theorem 3 holds with the choice
\begin{equation}
\label{1-13}
\tau = \min(1,\frac{\log^4U}{\log^3x}),
\end{equation}
so that
\begin{equation}
\label{1-14}
\Delta(x) \ll x^{1/2} \log^2U.
\end{equation}
Then, for example, the bound for $\Delta(x)$ in \cite{Li-Ye/2001} follows from \eqref{1-14} if the hypothesis in Theorem 3 holds with $U = \exp(\log^{5/8}x)$ and $\tau=1/\sqrt{\log x}$. Moreover, if such hypothesis holds with $U=\log\log x$ and $\tau=(\log\log\log x)^4/\log^3x$, then \eqref{1-14} gives
\[
\Delta(x) \ll x^{1/2} (\log\log\log x)^2,
\]
essentially Montgomery's conjectural upper bound for $\Delta(x)$ in \cite{Mon/1980}. Note that, apart from Heath-Brown \cite{HB/1982}, all these results require a very wide uniformity in $T$, entering the range $T\leq x^\varepsilon$; however, the required value of $\tau$ is not very small. In Theorem 1 we have the opposite situation, i.e. $T$ lies in a plausible range but we require a much wider uniformity in $\tau$.

{\bf 5.} In view of the classical $\Omega$-results in prime number theory, \eqref{1-13} and \eqref{1-14} show that there are definite limitations to the uniformity ranges of the bound $F(x,T,\tau)\ll T\log T$. A similar remark could be applied to Theorem 1 as well (once such results are refined by replacing $X^\varepsilon$ by the expected powers of $\log X$) in view of the oscillation results of Maier's \cite{Mai/1985} type for primes in short intervals. {\hfill \qedsymbol}

\medskip
We conclude observing that it would be desirable to have some numerical evidence towards bounds and uniformity ranges for $F(X,T,\tau)$, but such computations could be quite heavy.

\bigskip
{\bf Acknowledgements.}
This research was partially supported by the grant PRIN2010-11 {\sl Arithmetic Algebraic Geometry and Number Theory}.
The first named author was partially supported by the Cariparo
``Eccellenza'' grant \textsl{Differential Methods in Arithmetic,
Geometry and Algebra}.

\bigskip
\section{Proof of Theorem 1}

\smallskip
We assume from the beginning that $1 \leq h \leq X$ and $0\leq Y \leq X$. We start with the classical explicit formula, see Davenport \cite{Dav/2000}, in the form
\[
\psi (x) = x - \sum_{|\gamma|\le  X} \frac{x^\rho}{\rho} + O(\log^2X)
\]
uniformly for $X\leq x \leq 2X$, thus obtaining
\begin{equation}
\label{2-1}
J(X,Y,h) \ll \int_X^{X+Y} \big|\sum_{|\gamma|\le  X} \frac{(x+h)^\rho - x^\rho}{\rho}\big|^2 \mathrm{d} x+ YX^\varepsilon.
\end{equation}
Let $1\leq U\le  X$ be a parameter to be chosen later on. We first evaluate the contribution of the terms in \eqref{2-1} with $\vert \gamma \vert \le  U$. Since $\rho=1/2+i\gamma$ we have
\[
\sum_{\vert \gamma \vert\le  U}  \frac{(x+h)^\rho-x^\rho}{\rho} = \int_{x}^{x+h} \sum_{\vert \gamma \vert\le  U}  u^{\rho-1}\ \mathrm{d} u \ll x^{-1/2} \int_{x}^{x+h} \Bigl\vert \sum_{\vert \gamma \vert\le  U}  u^{i\gamma} \Bigr\vert\ \mathrm{d} u,
\]
hence by the Cauchy-Schwarz inequality we have
\begin{equation}
\label{2-2}
\int_{X}^{X+Y}  \Big \vert \sum_{\vert \gamma \vert\le  U} \frac{(x+h)^\rho-x^\rho}{\rho}\Big \vert^2\mathrm{d} x
\ll \frac{h}{X} \int_{X}^{X+Y} \Big( \int_{x}^{x+h} \Bigl\vert \sum_{\vert \gamma \vert\le  U}  u^{i\gamma}
\Bigr\vert^2\mathrm{d} u \Big) \mathrm{d} x.
\end{equation}
We deal with the right hand side of  \eqref{2-2} first changing the order of integration and then adapting part of the arguments in Heath-Brown \cite{HB/1982}.

\medskip
{\bf Lemma 1.} {\sl For $Y,h\geq 0$ and a continuous function $f(u)$ we have}
\[
\int_X^{X+Y}\Big(\int_x^{x+h} f(u)\mathrm{d} u\Big)\mathrm{d} x = \int_X^{X+h}\Big(\int_x^{x+Y} f(u)\mathrm{d} u\Big)\mathrm{d} x.
\]

\medskip
{\it Proof.} Let $x\geq X$ and $g(x) = \int_X^x f(u)\mathrm{d} u$. Then the left hand side is
\[
\begin{split}
\int_X^{X+Y} \big(g(x+h)&-g(x)\big)\mathrm{d} x = \int_{X+h}^{X+Y+h} g(x)\mathrm{d} x - \int_{X}^{X+Y} g(x)\mathrm{d} x \\
&= \int_{X+Y}^{X+Y+h} g(x)\mathrm{d} x - \int_{X}^{X+h} g(x)\mathrm{d} x = \int_X^{X+h} \big(g(x+Y)-g(x)\big)\mathrm{d} x
\end{split}
\]
and the lemma follows. {\hfill \qedsymbol}

\medskip
We extend the definition of $\Sigma(X, T)$ in \eqref{1-2} writing for $v\in\mathbb{R}$
\[
\Sigma(X,T;v) =  \sum\limits_{|\gamma|  \le  T} X^{i\gamma} e^{i\gamma v}.
\]

\medskip
{\bf Lemma 2.} {\sl  For $V,T \geq1$ and $\tau\in(0,1]$ we have}
\[
\int_{\mathbb{R}} \vert \Sigma(V,T;v) \vert^2 e^{-2 \vert v \vert /\tau} \mathrm{d} v = \tau F(V,T,\tau).
\]

\medskip
{\it Proof.} This is a variation on Lemma 3 of \cite{HB/1982}. Squaring out and using the substitution $v=y\tau$, the left hand side equals
\[
\tau \sum_{-T \le  \gamma , \gamma'\le  T} V^{i(\gamma - \gamma')} \int_{\mathbb{R}} e^{iy\tau(\gamma - \gamma')}
e^{-2 \vert y \vert } \mathrm{d} y = \tau F(V,T,\tau),
\]
thanks to the Fourier transform formula
\[
\int_{\mathbb{R}} e^{iy\tau (\gamma - \gamma')} e^{-2 \vert y \vert } \mathrm{d} y = \frac{4}{4+\tau^2(\gamma-\gamma')^2};
\]
see Lemma 3 of \cite{HB/1982}. {\hfill \qedsymbol}

\medskip
{\bf Lemma 3.} {\sl For $V,T \geq1$ and $\tau\in[0,1]$ we have}
\[
\int_{V}^{V(1+\tau)} \Bigl\vert  \sum_{|\gamma|\le  T} u^{i\gamma} \Bigr\vert^2  \mathrm{d} u \ll V \tau F(V,T,\tau).
\]

\medskip
{\it Proof.} The case $\tau=0$ is trivial. By the substitution $u=Ve^{v}$, and observing that $e^{v} \le  e^3 e^{-2v/\tau}$ for $\tau\in (0,1]$ and $v\in [0,\log(1+\tau)]$, similarly as in (9) of \cite{HB/1982} we get that the left hand side equals
\[
\begin{split}
V \int_{0}^{\log(1+\tau)}  \vert \Sigma(V,T;v) \vert^2 e^{v}\mathrm{d} v &\ll V \int_{0}^{\log(1+\tau)}  \vert \Sigma(V,T;v) \vert^2 e^{-2v/\tau}\mathrm{d} v \\
&\ll  V\int_{\mathbb{R}}  \vert \Sigma(V,T;v) \vert^2 e^{- 2\vert v \vert/ \tau}\mathrm{d} v =  V \tau F(V,T,\tau),
\end{split}
\]
thanks to Lemma 2. {\hfill \qedsymbol}

\medskip
Changing the order of integration in \eqref{2-2} by Lemma 1, and then applying Lemma 3 with $V=x, T=U$ and $\tau=Y/x$, from \eqref{2-2} we obtain
\begin{equation}
\label{2-3}
\int_{X}^{X+Y}  \Big \vert \sum_{\vert \gamma \vert\le  U} \frac{(x+h)^\rho-x^\rho}{\rho}\Big \vert^2\mathrm{d} x \ll  \frac{h^2Y}{X} \max_{X\le  x \le  X+h} F(x, U, \frac{Y}{x}).
\end{equation}

\medskip
Next we consider the contribution from the terms $U<\vert \gamma \vert \le  X$. We need a further lemma.

\medskip
{\bf Lemma 4.} {\sl For $V\geq 1$ and $\tau\in[0,1]$ we have}
\[
\int_{V}^{V(1+\tau)} \Bigl \vert \sum_{U< \vert \gamma \vert \le  X}  \frac{u^{i\gamma}}{\rho} \Bigr\vert^2 \mathrm{d} u \ll
V \tau \Big(\frac{F(V,X,\tau)}{X^2} + \frac{F(V,U,\tau)}{U^2} + \frac{1}{U^{1/2}}  \int_U^X F(V,t,\tau) \frac{\mathrm{d} t}{t^{5/2}}\Big).
\]

\medskip
{\it Proof.} Analogously to p.93-94 of \cite{HB/1982}, applying partial summation to the inner sum, using the Cauchy-Schwarz inequality and applying Lemma 3 with $U\leq T\leq X$ we have
\begin{align*}
\int_{V}^{V(1+\tau)} & \Bigl \vert \sum_{U<\vert \gamma \vert\le X} \frac{u^{i\gamma}}{\rho} \Bigr \vert^2 \mathrm{d} u
\ll  \frac{1}{X^2} \int_{V}^{V(1+\tau)} \Bigl \vert \sum_{U<\vert \gamma \vert\le X} u^{i\gamma} \Bigr \vert^2 \mathrm{d} u
+ \int_{V}^{V(1+\tau)} \Bigl ( \int_U^X \Bigl \vert \sum_{U<\vert \gamma \vert\le  t} u^{i\gamma} \Bigr \vert  \frac{\mathrm{d} t}{t^2}
\Bigr )^{2} \mathrm{d} u \\
&\ll \frac{V \tau}{X^2} \Bigl(F(V,X,\tau) + F(V,U,\tau) \Bigr) + \int_{V}^{V(1+\tau)} \Big(\int_U^X \Big \vert \sum_{U<\vert \gamma \vert\le t}  u^{i\gamma} \Big \vert^2 \frac{\mathrm{d} t}{t^{5/2}}\Big) \Big(\int_U^X \frac{\mathrm{d} t}{t^{3/2}} \Big) \mathrm{d} u \\
&\ll \frac{V \tau}{X^2} \Bigl( F(V,X,\tau) + F(V,U,\tau) \Bigr) + \frac{V \tau}{U^{1/2}}\int_U^X \Bigl(F(V,t,\tau) + F(V,U,\tau)\Bigr) \frac{\mathrm{d} t}{t^{5/2}} \\
&\ll V \tau\Bigl(\frac{F(V,X,\tau)}{X^2} + \frac{F(V,U,\tau)}{U^2} +\frac{1}{U^{1/2}} \int_U^X F(V,t,\tau)\frac{\mathrm{d} t}{t^{5/2}}\Bigr),
\end{align*}
which proves the lemma. {\hfill \qedsymbol}

\medskip
Writing
\[
\Bigl\vert\sum_{U<\vert \gamma \vert\le  X} \frac{(x+h)^\rho-x^\rho}{\rho}\Bigr\vert^2 \ll \Bigl \vert \sum_{U<\vert \gamma \vert\le  X} \frac{(x+h)^\rho}{\rho} \Bigr \vert ^2+ \Bigl \vert \sum_{U<\vert \gamma \vert\le X} \frac{x^\rho}{\rho} \Bigr \vert^2,
\]
applying Lemma 4 with $V=X+h$, $\tau=Y/(X+h)$ and with $V=X$, $\tau = Y/X$ we obtain
\begin{equation}
\label{2-4}
\begin{split}
&\int_{X}^{X+Y} \Big \vert \sum_{U<\vert \gamma \vert\le X}\frac{(x+h)^\rho-x^\rho}{\rho}\Big \vert^2\mathrm{d} x \ll \\
& \hskip.5cm YX \Bigl\{ \frac{F(X+h,X,\frac{Y}{X+h})}{X^2} + \frac{F(X+h,U,\frac{Y}{X+h})}{U^2}  + \frac{1}{U^{1/2}}\int_{U}^{X} F\big(X+h,t,\frac{Y}{X+h}\big) \frac{\mathrm{d} t}{t^{5/2}} \\
&\hskip1.5cm + \frac{F(X,X,\frac{Y}{X})}{X^2} + \frac{F(X,U,\frac{Y}{X})}{U^2} + \frac{1}{U^{1/2}}\int_{U}^{X} F\big(X,t,\frac{Y}{X}\big) \frac{\mathrm{d} t}{t^{5/2}} \Bigr\}.
\end{split}
\end{equation}

\medskip
Assume now hypothesis $H(\eta)$ with $0<\eta<1$. Since we use the bound $F(X,T,\tau) \ll TX^\varepsilon$ in the range $U\leq T \leq X$, from  \eqref{2-1}, \eqref{2-3} and \eqref{2-4} we may write
\begin{equation}
\label{2-5}
\begin{split}
J(X,Y,h) &\ll  \frac{h^2Y}{X}F(X,U,\frac{Y}{X}) + YX \frac{F(X,U,\frac{Y}{X})}{U^2} + YX^\varepsilon \\
&\ll  \frac{h^2YUX^\varepsilon}{X} +\frac{YX^{1+\varepsilon}}{U} + YX^\varepsilon
\end{split}
\end{equation}
under the conditions imposed by hypothesis $H(\eta)$, namely
\begin{equation}
\label{2-6}
X^\eta \leq U \leq X \  \ \text{and} \ \ \frac{X^\eta}{U} \leq \frac{Y}{X} \leq 1.
\end{equation}
The optimal choice of $U$ in \eqref{2-5} is
\begin{equation}
\label{2-7}
U=\frac{X}{h},
\end{equation}
thus getting the required bound for $J(X,Y,h)$ in \eqref{1-12}. Moreover, \eqref{2-6} and \eqref{2-7} give the uniformity conditions on $h$ and $Y$ in the first part of Theorem 1. When $\eta=0$ we require
\begin{equation}
\label{2-8}
X^\varepsilon \leq U \leq X \  \ \text{and} \ \ 0 \leq \frac{Y}{X} \leq 1,
\end{equation}
therefore \eqref{2-7} and \eqref{2-8} give the uniformity conditions in the second part of Theorem 1.

\bigskip
\section{Comparing $F(X,T,\tau)$ and $\Sigma(X,T)$}

\smallskip
In this section we link $F(X,T,\tau)$ and $\Sigma(X,T)$, as required in Remark 1 of Section 1 and by the applications of $F(X,T,\tau)$ in Theorem 2.

\medskip
{\bf Lemma 5.} {\sl Let $\varepsilon>0$, $X^\varepsilon\leq T \leq X$ and $\tau T \leq  X^\varepsilon$. Then}
\[
F(X,T,\tau) \ll T^{1+\varepsilon}  + X^{\varepsilon} \max_{t\leq T} \vert \Sigma(X,t) \vert^2.
\]

\medskip
{\it Proof.}  For $\tau=0$ the result is trivial, so we assume that $\tau>0$. Writing $\delta=(\tau/2)(1-\varepsilon/2) \log T$ and recalling the definition of $\Sigma(X,T;v)$ after Lemma 1, thanks to the trivial bound $\Sigma(X,T;v)  \ll  T\log T$ we get
\[
\Bigl(\int_{-\infty}^{-\delta}+ \int_{\delta}^{+\infty} \Bigr) \vert \Sigma(X,T;v) \vert^2e^{-2 \vert v \vert /\tau}\mathrm{d} v \ll
\tau T^{1+\varepsilon}.
\]
Hence in view of Lemma 2 with $V=X$ we obtain
\[
\tau F(X,T,\tau) = \int_{-\delta}^{\delta} \vert \Sigma(X,T;v) \vert^2 e^{-2 \vert v \vert /\tau} \mathrm{d} v +O(\tau T^{1+\varepsilon}).
\]
By partial summation we have
\[
\Sigma(X,T;v) =e^{ivT}\sum_{|\gamma|\le  T}X^{i\gamma}-iv \int_{0}^{T}\big( \sum_{\vert\gamma\vert\le  t} X^{i\gamma} \big) e^{ivt} \mathrm{d} t\ll(1+\vert v \vert T)\max_{t\leq T}  \vert \Sigma(X,t) \vert,
\]
hence thanks to the hypothesis $\tau T \leq  X^\varepsilon$ we get
\[
\begin{split}
\int_{-\delta}^{\delta} \vert \Sigma(X,T;v) \vert^2 e^{-2 \vert v \vert /\tau} \mathrm{d} v &\ll \tau \log T (1+\tau^2 T^2 \log^2 T)
\max_{t\leq T}  \vert \Sigma(X,t) \vert^2 \\
&\ll \tau X^{\varepsilon}
\max_{t\leq T} \vert \Sigma(X,t) \vert^2.
\end{split}
\]
Lemma 5 is therefore proved. {\hfill \qedsymbol}

\medskip
In the opposite direction, since $F(X,T,0)=|\Sigma(X,T)|^2$ we have that hypothesis $H(0)$ implies Gonek's conjecture in the range $X^\varepsilon\leq T \leq X$. We expect that a weaker hypothesis, namely the bound \eqref{1-10} uniformly for $X^\varepsilon\leq T \leq X$ and $1/T\leq \tau\leq 1$, should already imply Gonek's conjecture in the range $X^\varepsilon\leq T \leq X$. However, all we can prove at present is the following modified form of Lemma 1 of Heath-Brown $\&$ Goldston \cite{HB-Go/1984}. Writing
\begin{equation}
\label{3-1}
\Sigma(X,U,T)=\sum_{U<\vert \gamma \vert \leq T}  X^{i\gamma}
\end{equation}
and observing that $F(X,T,\tau)\geq 0$ thanks to Lemma 2, we have

\medskip
{\bf Lemma 6.} {\sl For $X,T\geq 2$, $0\leq U < T$ and $0\leq \tau\leq 1$ we have}
\[
\Sigma(X,U,T) \ll (1+T\tau)^{1/2}\, \max_{U\leq t\le  T} F(X,t,\tau)^{1/2}.
\]

\medskip
{\it Proof.} The case $\tau=0$ is trivial since $\vert \Sigma(X,0,T) \vert^2 = F(X,T,0)$, hence
\[
\vert \Sigma(X,U,T) \vert \leq \vert \Sigma(X,0,T) \vert + \vert \Sigma(X,0,U)\vert \ll \max_{U\leq t\le  T} F(X,t,0)^{1/2}.
\]
For $\tau>0$ we write
\[
\Sigma(X,U,T;v) =\sum_{U< \vert \gamma \vert \leq T}  X^{i\gamma} e^{iv\gamma}
\]
and apply the Sobolev-Gallagher inequality
\[
\vert f(0) \vert \leq \frac{1}{2\tau} \int_{-\tau}^{\tau}\vert f(v) \vert \mathrm{d} v + \frac12  \int_{-\tau}^{\tau}\vert f'(v) \vert \mathrm{d} v
\]
with $f(v)=\Sigma(X,U,T;v)^{2}$, thus obtaining
\begin{equation}
\label{3-2}
\vert \Sigma(X,U,T)\vert^{2} \ll \frac{1}{\tau} \int_{-\tau}^{\tau}\vert \Sigma(X,U,T;v) \vert^{2}\mathrm{d} v +  \int_{-\tau}^{\tau}
\vert \Sigma(X,U,T;v) \vert \Bigl\vert \frac{\partial}{\partial v} \Sigma(X,U,T;v) \Bigr\vert  \mathrm{d} v.
\end{equation}
By partial summation we have
\[
\Bigl\vert \frac{\partial}{\partial v} \Sigma(X,U,T;v) \Bigr\vert = \Bigl\vert \sum_{U<\vert \gamma \vert \leq T} \gamma X^{i\gamma} e^{iv\gamma} \Bigr\vert  \leq T \vert \Sigma(X,U,T;v) \vert + \Bigl\vert \int_{U}^{T} \Sigma(X,U,t;v) \mathrm{d} t \Bigr\vert,
\]
hence by the Cauchy-Schwarz inequality we get
\begin{equation}
\label{3-3}
 \int_{-\tau}^{\tau} \Bigl\vert \frac{\partial}{\partial v} \Sigma(X,U,T;v) \Bigr\vert^{2} \mathrm{d} v
\ll T ^{2} \int_{-\tau}^{\tau}\vert\Sigma(X,U,T;v)\vert^{2}\mathrm{d} v+T^2\max_{U\leq t\leq T} \int_{-\tau}^{\tau}\vert\Sigma(X,U,t;v)\vert^{2} \mathrm{d} v.
\end{equation}
But, recalling that $\Sigma(X,0,T;v)=\Sigma(X,T;v)$, for $U\leq t \leq T$ Lemma 2 gives
\begin{equation}
\label{3-4}
\begin{split}
\int_{-\tau}^{\tau} \vert\Sigma(X,U,t;v)\vert^{2}\mathrm{d} v&\ll \int_{-\infty}^{\infty} \left(\vert\Sigma(X,0,t;v)\vert^{2} +  \vert\Sigma(X,0,U;v)\vert^{2}\right) e^{-2\vert v \vert/\tau} \mathrm{d} v \\
&\ll \tau \max_{U\leq t\leq T} F(X,t,\tau),
\end{split}
\end{equation}
therefore Lemma 6 follows by the Cauchy-Schwarz inequality from \eqref{3-2}, \eqref{3-3} and \eqref{3-4}. {\hfill \qedsymbol}

\bigskip
\section{Proof of Theorem 2}

\smallskip
Let $\varepsilon>0$ be arbitrarily small; in what follows the dependence on $\varepsilon$ of the ranges of $h$ and $\eta$ will be written in a slightly weaker but cleaner form. We first show that the estimate for $J(X,Y,h)$ in Theorem 1, under $H(\eta)$ with $\eta>0$, implies
\begin{equation}
\label{4-1}
\psi(x+h)-\psi(x)=h+O(h^{2/3}x^{\eta/3+\varepsilon}).
\end{equation}
Moreover, \eqref{4-1} is non-trivial (in the sense of Remark 2) for $x^{\eta + 5\varepsilon} \leq h \leq x^{3/4 - \eta/2}$ and $ 0 < \eta < 1/2 - 4\varepsilon$. The case $\eta=0$ will be treated by a direct application of $H(0)$, since we already observed in Remark 2 that both approaches give the same result in this case. We start noticing that if there exist $x_0\in[X,X+Y]$ and $\log X \leq K=o(h)$ such that $|\psi(x_0+h)-\psi(x_0)-h|\geq 5K$, then
\begin{equation}
\label{4-2}
|\psi(x+h)-\psi(x)-h|\geq K \ \ \text{for every} \ \ x\in[X,X+Y]\cap \left[x_0-\frac{K}{\log X},x_0+\frac{K}{\log X}\right].
\end{equation}
Indeed, clearly both $\psi(x+h)$ and $\psi(x)$ may change at most by $\pm K$ with respect to their values at $x=x_0$, as $x$ runs over such an interval. Assuming that the above inequality holds for some $x_0\in[X,X+Y]$ with $K=h^a$ and $0<a<1$, from Theorem 1 with the choice $Y=hX^\eta$ we get
\begin{equation}
\label{4-3}
\frac{h^{3a}}{\log X} \ll J(X,Y,h) \ll h^2X^{\eta+\varepsilon},
\end{equation}
since the cardinality of the $x$ in \eqref{4-2} is $\gg h^a/\log X$. Hence \eqref{4-1} follows, since \eqref{4-3} is contradictory if $h^a> ch^{2/3}X^{\eta/3+\varepsilon}$ with a certain $c>0$, say. Actually, this argument gives \eqref{4-1} in a wider range of $h$, but a computation shows that the bound is non-trivial essentially only in the stated range.

\medskip
Next we show that a direct application of $H(\eta)$ gives
\begin{equation}
\label{4-4}
\psi(x+h)-\psi(x) - h \ll
\begin{cases}
h^{1/3} x^{1/6+\eta/3+\varepsilon} & \text{if} \ 0<\eta<1 \\
h^{1/2} x^{\varepsilon} & \text{if} \ \eta=0;
\end{cases}
\end{equation}
\eqref{4-4} is non-trivial (again in the sense of Remark 2) for  $x^{1/4 + \eta/2 + 3 \varepsilon} \leq h \leq x^{1 - \eta}$ if $0 < \eta < 1/2 - 2 \varepsilon$ and for $x^{3 \varepsilon} \leq h \leq x^{1 - \varepsilon}$ if $\eta=0$. The proof of \eqref{4-4} is along the lines of Theorem 4 in Gonek \cite{Gon/1993}, especially in the case $\eta=0$. By the explicit formula we have
\begin{equation}
\label{4-5}
\psi(x+h) - \psi(x) = h - \sum_{|\gamma|\leq x/h} \int_x^{x+h}u^{\rho-1}\mathrm{d} u  - \sum_{x/h < \vert \gamma \vert \leq x}\frac{(x+h)^\rho-x^{\rho}}{\rho} + O(\log^{2} x).
\end{equation}
Assume first that $0<\eta<1$ and recall \eqref{1-2}, \eqref{3-1} and the trivial bound $\Sigma(X,T)\ll T\log T$. Hence, applying Lemma 6 with $x\leq X\leq x+h$, $x^\eta\leq U\leq x/h$, $T=x/h$ and $\tau=x^\eta/U$, from hypothesis $H(\eta)$ we obtain
\begin{equation}
\label{4-6}
\begin{split}
\sum_{|\gamma|\leq x/h} \int_{x}^{x+h} u ^{\rho-1} \mathrm{d} u &\ll hx^{-1/2}\big(\max_{x\leq u\leq x+h}|\Sigma(u,U)| + \max_{x\leq u\leq x+h}|\Sigma(u,U,\frac{x}{h})|\big) \\
&\ll hx^{-1/2+\varepsilon}(U + \frac{x^{1+\eta/2}}{hU^{1/2}})  \ll h^{1/3} x^{1/6 + \eta/3 + \varepsilon}
\end{split}
\end{equation}
with the optimal choice $U=(x^{2+\eta}/h^2)^{1/3}$, provided $x^\eta\leq x/h\leq x$. Note that the above conditions on $U$ are satisfied with such a choice. Let now $u=x$ or $u=x+h$. By partial summation and Lemma 6 with $X=u$, $U=x/h$, $x/h\leq T \leq x$ and $\tau =hx^{\eta-1}$, from hypothesis $H(\eta)$ we get
\begin{equation}
\label{4-7}
\sum_{x/h < |\gamma| \leq x} \frac{u^\rho}{\rho} \ll x^{-1/2} \vert \Sigma(u,\frac{x}{h},x) \vert
+ x^{1/2} \int_{x/h}^{x} \vert \Sigma(u,\frac{x}{h},t)\vert \frac{\mathrm{d} t}{t^2} \ll h^{1/2} x^{\eta/2+\varepsilon},
\end{equation}
again provided  $x^\eta\leq x/h\leq x$. From \eqref{4-6} and \eqref{4-7} we see that \eqref{4-5} becomes
\[
\psi(x+h) - \psi(x) - h \ll (h^{1/3} x^{1/6 + \eta/3} + h^{1/2} x^{\eta/2})x^\varepsilon.
\]
A simple computation shows that the term $h^{1/3} x^{1/6 + \eta/3}$ dominates if $h\leq x^{1-\eta}$ and is non-trivial in the range stated after \eqref{4-4}. Moreover, the term $ h^{1/2} x^{\eta/2}$ dominates if $h\geq x^{1-\eta}$, but gives a trivial result in such a range. Thus the bound \eqref{4-4} follows if $\eta>0$. If $\eta=0$ the argument is similar but simpler, and reduces essentially to the proof of Theorem 4 of Gonek \cite{Gon/1993}. Indeed, the freedom in the choice of $\tau$ given by hypothesis $H(0)$ allows to choose $U=x^\varepsilon$ and $\tau=0$ in the application of Lemma 6 to the first row of \eqref{4-6}, thus giving
\begin{equation}
\label{4-8}
 \sum_{|\gamma|\leq x/h} \int_x^{x+h}u^{\rho-1}\mathrm{d} u \ll h^{1/2}x^\varepsilon,
\end{equation}
provided $x^{3 \varepsilon} \leq h \leq x^{1 - \varepsilon}$. Hence the bound \eqref{4-4} follows from \eqref{4-8} and \eqref{4-7} in the case $\eta=0$, and is non-trivial in the stated range. Theorem 2 follows from \eqref{4-1} and \eqref{4-4}.

\vskip2cm

redpajama_set_name: RedPajamaArXiv


text:
Amy Shuckburgh has produced another exciting collection of original artwork for exhibition at Picture. Following the success of her show with the gallery last year, 26 new landscapes, as well as figurative and still life works, have arrived at the gallery and will be framed in two different styles, available from mid-May 2022.
Amy's bold use of soft pastels conveys her confident feel for colour. She often works en plein air, as well as developing larger works back at her studio. Amy's new collection sees a return to some of her favourite places: the Cornish coast, the pretty villages of the Ile de Re in France, as well as the woods, hills and rivers around her home in Somerset.
She has been praised for the honesty and immediacy of her depictions of motherhood and her series celebrating the maternal relationship – some of which are included in this collection – won her the Heatherley's Award for Drawing in 2019. Amy is also known for her portraiture. Her large-scale drawing of the late playwright Harold Pinter, done from sittings just before his death, is owned by Pinter's widow, Antonia Fraser, and now hangs permanently at the Harold Pinter Theatre in London.
'I love Amy Shuckburgh's work. She has a lyrical sense of colour, which coupled with her evident passion for the landscape, make this show instantly engaging,'
Lady Antonia Fraser
Home / Artists / Amy Shuckburgh
Artist's Farm
Blue hill beyond
Bluebells & Wild Garlic
Boat Yard
Bruton Hills
Cliff Path
Downhill from Cadbury Castle
French Siesta shadows, La Couarde, Ile de Re
Harbour at Night
High Lane, Summer
High Summer Island