Fernando Solano was recently honored with the 2016 Teacher of the Year award for his work as an English as a Second Language (ESL) teacher. The South Lexington Elementary School teacher accepted the award while several other teachers were given recognition as "Superheroes in the Classroom".
Solano said that the award came as a big surprise as well as an honor. "No teacher could be happier than when we are recognized for what we do," he said.
Jennifer Brown, the principal at the elementary school, described Solano as a dedicated and passionate teacher who provided a thorough experience to the students. She said that Solano is connected to both the student's and the parent's needs when it comes to literacy. For his part, Solano infuses his love of literacy into his students. "He has a vested interest in ESL and working with the children," said Brown.
This is Solano's second year with South Lexington, having previously worked in Guilford County schools.
Rick Kriesky, Superintendent for the Lexington City School district, admitted choosing the Teacher of the Year was a challenge because he believed all teachers deserve to be honored.
Emy Garret, assistant superintendent concurred describing the teachers as superheroes who have superhuman qualities including multi-tasking, instilling a passion to learn and keeping peace in the classrooms.
During the awards, Kriesky complimented all of the teachers during the annual brunch to celebrate the Teacher of the Year. He went on to congratulate them for doing an amazing job inspiring the children in their classrooms and beyond.
Several other teachers were given recognition for individual school teacher of the year during the brunch. In 2015, the Teacher of the Year award was shared by two teachers for the first time in its history.


text:
Мойсей Маркович Альперін (1881–1951) — український лікар–фтизіатр, професор, доктор медичних наук.

Біографія
Мойсей Маркович Альперін народився в 1881 році  у Білгородці Волинської губернії. З 1892 року мешкав в Одесі.

У 1911 році закінчив медичний факультет Новоросійського університету. У 30 років почав лікарську приватну практику. В 1920—1922 роках був ординатором військового шпиталю, а з 1924 року — ординатором туберкульозного інституту. Все подальше життя було пов'язано з фтизіатрією.

В 1934 році став доцентом інституту вдосконалення лікарів, а в 1935 році — кандидатом медичних наук. Керував роботою аспірантів та пошукувачів.

В 1941 році в Києві захистив дисертацію з проблем туберкульозу тракту травлення на здобуття наукового ступеня доктора медичних наук. Згодом було присвоєно вчене звання професора. Читав лекції в медичному інституті.

В роки нацистської навали очолював клініку Науково-дослідного інституту туберкульозу в Ташкенті. Після повернення до Одеси працював у протитуберкульозному диспансері.

Помер в 1951 році в м. Одеса.

Праці
Є автором опублікованих наукових статей та монографії.

 Туберкулез пищеварительного тракта: Клиника, диагностика, терапия/ М. М. Альперин. — Киев, 1950.

Оцінка діяльності
Найбільш вдала систематизація туберкульозу кишечнику — класифікація М. М. Альперіна, в котрій виділені безсимптомна, діарейна, алергічна, диспепсична та загально інтоксикаційна форми захворювання. Загальнопризнаною є клініко-морфологічна класифікація туберкульозного перитоніту, який включає бугоркову, ексудативну, злипчиву, ексудативно-злипчиву казеозно-виразкову форми. (Катеньов Валентин Львович, лікар-рентгенолог, https://radiomed.ru/publications/12369-tuberkulyoz-abdominalnyy-tuberkulez )

Родина
 Син: Марк Мойсейович Альперін (1916—1994) — фізик-теоретик, професор, завідувач кафедри фізики Одеського педагогічного інституту ім. К. Д. Ушинського

Посилання
 Альперін М. М.// Кто есть кто// https://www.migdal.org.ua/times/42/3659/

Радянські медики
Фтизіатри
Науковці Одеси
Радянські професори

redpajama_set_name: RedPajamaWikipedia


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"To run from vulnerability is to run from the essence of our nature, the attempt to be invulnerable is the vain attempt to become something we are not..." says the poet David Whyte. In her New York Times best-selling book Daring Greatly, Brené Brown makes an in-depth exploration of vulnerability, how we are taught to see it as weakness and how we are conditioned to use shame and self-judgment to try to make it go away. She says that in reality, allowing ourselves to be vulnerable is intimately connected with strength and courage. It transforms the way we live, love, parent and lead. In this experiential workshop we will use meditation, inquiry and specific exercises to discover our own relationship to vulnerability - how we defend against it and also the way in which vulnerability shows up in our lives as an ally and an essential aspect of our true nature.
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Teacher Bio: Bruce Schneider, Doctor of Chiropractic has been in private practice for 25 years. He is a graduate of New York Chiropractic College and has trained with the Upledger Institute for Cranio-Sacral Therapy. Dr. Schneider completed a four year program in the Pathwork, a dynamic process for deepening self awareness.


We all know the importance of making a "good first impression" in face-to-face meetings, but many companies fail to remember that this applies to email interactions as well. A significant amount of your future email marketing success will rely on your subscribers liking what they see in those initial emails you send to them. Here's how to get your email campaigns off to a solid start.
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What is inbound marketing? In this guide, we're stripping it back to basics and going deep into the value of inbound marketing, its elements and specifics. We're also bringing you an ultimale glossary of the key inbound marketing terms!
Hungry for inbound marketing? Yum yum! We take a look at the importance of personal branding, social listening and email marketing in this week's edition of Friday Night Takeaway. All of the week's best news, tips, tactics and strategies.


text:
49ers have favorable schedule for 2021 NFL season
July 15, 2021 by Niners Nation Leave a Comment
Photo by Thearon W. Henderson/Getty Images
San Francisco will play a last-place schedule after finishing in the basement of the NFC West last season.
The 2021 NFL season officially kicks off on September 9 when the defending champion Tampa Bay Buccaneers host the Dallas Cowboys.
The San Francisco 49ers will begin their quest for a sixth Lombardi Trophy three days later when they're on the road to take on the Detroit Lions. Oddsmakers like the moves general manager John Lynch and head coach Kyle Shanahan made this offseason, giving the Niners the second-best odds to win the NFC.
With Nick Bosa, George Kittle, Fred Warner and Trent Williams headlining the roster, San Francisco has high hopes going into next season.
Of course, expectations are similar to what they were going into the 2020 campaign. The 49ers were coming off a devastating loss in the Super Bowl and fans were hoping the team would go on a revenge tour to capture the title.
Then, Week 2 happened. The Niners lost Bosa, Kittle, Jimmy Garoppolo, Richard Sherman and Raheem Mostert to long-term injuries by the end of their second game and never recovered, finishing last season with a 6-10 record.
There is a big difference between the past two years when it comes to the 49ers. Coming off a 13-3 finish in 2019, the Niners played a first-place schedule in 2020. They finished in the basement of the NFC last season, which means the schedule will be more favorable in 2021.
Here is the 49ers' full 2021 schedule:
Week 1: @ Lions (10:00 a.m.)
Week 2: @ Philadelphia Eagles (10:00 a.m)
Week 3: vs. Green Bay Packers (5:20 p.m. – SNF)
Week 4: vs. Seattle Seahawks (1:05 p.m.)
Week 5: @ Arizona Cardinals (1:25 p.m.)
Week 6: BYE
Week 7: vs. Indianapolis Colts (5:20 p.m. – SNF)
Week 8: @ Chicago Bears (10 a.m.)
Week 9: vs. Cardinals (1:05 p.m.)
Week 10: vs. Los Angeles Rams (5:15 p.m. – MNF)
Week 11: @ Jacksonville Jaguars (10:00 a.m.)
Week 12: vs. Minnesota Vikings (1:25 p.m.)
Week 13:@ Seahawks (5:20 p.m. – SNF)
Week 14: @ Cincinnati Bengals (10:00 a.m.)
Week 15: vs. Atlanta Falcons (1:05 p.m.)
Week 16: @ Tennessee Titans (5:20 p.m. – TNF)
Week 17: vs. Houston Texans (1:05 p.m.)
Week 18: @ Rams (1:25 p.m.)
Their opponents combined to go 132-138-2 in 2020, good for a .489 winning percentage, which is tied for the 13th easiest, according to the NFL Network. If you go by Vegas projected win totals, the 49ers have the easiest schedule in the league.
Because San Francisco finished fourth in the NFC West, it will have matchups against the Lions, Eagles and Falcons, the three teams who were last in the remaining three divisions in the conference in 2020.
The 2021 campaign marks the first time in history that we will have an 18-week regular season. Because of the last-place finish, the Niners will benefit from the extra game, taking on the rebuilding Cincinnati Bengals.
San Francisco played a fourth-place schedule two seasons ago and we all saw the result. With a healthy Garropolo and Bosa in the lineup plus more overall talent on the roster than in 2018, the 49ers rolled to an 8-0 start en route to winning the division.
Shanahan is hoping for a similar start to the 2021 campaign. San Francisco begins the schedule with back-to-back games on the east coast. As we all know, nothing is guaranteed in the NFL, but both Detroit and Philadelphia aren't expected to be very good next season.
Similar to past years when they played consecutive games in the east, the 49ers will likely spend the week between Games 1 and 2 in Youngstown, Ohio. This will give the players a chance to mesh and do some team-building during that time, just like they did between their wins against the Buccaneers and Bengals to begin the 2019 campaign.
The toughest stretch of the schedule comes early in the season. The Niners host the Packers and Seahawks in Weeks 3 and 4 before hitting the road for a matchup against the Cardinals. They'll get an early BYE in Week 6 before taking on DeForest Buckner and the Colts in Week 7.
There's no doubt the NFC West will be a blood bath in 2021. All four teams have playoff aspirations, and oddsmakers believe San Francisco, L.A. and Seattle are among the favorites to win the conference.

redpajama_set_name: RedPajamaCommonCrawl


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Wherever you may be in the world we are standing by to offer technical support, the procurement of parts, or to source quality contractors to assist you in regions where we may not have a physical presence.


I have pulled out my main SBIG CCD cmaera setup from C14 telescope now. The only reason is the heavy light pollution from Khyaban e Iqbal road.
ZWO ASI 174 planetary camera is back in C14 now and these day i am taking high resolution lunar images.


text:
New Rule Clarifies Religious Liberty Protections for Federal Contractors
The U.S. Dept. of Labor announced a proposed rule to ensure conscience and religious freedom for religious organizations who contract with the government.
Featured Sermon: Wisdom (James 3:13-18)
religious liberty,
The U.S. Department of Labor announced on Wednesday a proposed rule to ensure that conscience and religious freedom for religious organizations who contract with the federal government is given the broadest protection permitted by law.
The Department's press release cites federal law, Supreme Court decisions, and previous executive orders as the basis for this proposed rule:
The Civil Rights Act of 1964 includes a critical, protective exemption for religious organizations. A similar exemption is included in Executive Order 11246 and OFCCP's regulations, which govern certain employment practices of federal contractors. Recent Supreme Court decisions – Masterpiece Cakeshop v. Colorado Civil Rights Commission, Trinity Lutheran Church v. Comer, and Burwell v. Hobby Lobby Stores – further address the protections afforded religious organizations and individuals under the Constitution and federal law. Executive Orders 13798, Promoting Free Speech and Religious Liberty, and 13831, Establishment of a White House Faith and Opportunity Initiative, along with U.S. Department of Justice guidance, likewise instruct federal agencies to protect religious exercise and not impede it.
"(Wednesday's) proposed rule helps to ensure the civil rights of religious employers are protected," Acting U.S. Secretary of Labor Patrick Pizzella remarked. "As people of faith with deeply held religious beliefs are making decisions on whether to participate in federal contracting, they deserve clear understanding of their obligations and protections under the law."
Office of Federal Contract Compliance Programs Director Craig Leen said, "OFCCP is consistently looking for ways to bring clarity and certainty to federal contractors, and this proposal falls squarely within that effort. The rulemaking process allows the public opportunity to comment on the proposal and impact any potential final rule."
The proposed rule would clarify that religious organizations may make employment decisions consistent with their sincerely held religious tenets and beliefs without fear of sanction by the federal government. The proposal also reaffirms employers' obligations not to discriminate on the basis of race, sex, or other protected bases and does not exempt or excuse a contractor from complying with any other requirements.
Specifically, the proposed rule would do the following:
The term "religion" would be defined to provide that the term is not limited to religious belief but also includes all aspects of religious observance and practice.
To define "particular religion" to clarify that the religious exemption allows religious contractors not only to prefer in employment individuals who share their religion but also to condition employment on acceptance of or adherence to religious tenets as understood by the employing contractor. This would include all aspects of religious belief, observance, and practice as understood by the employer.
To receive the religious exemption a federal contractor must be organized for a religious purpose, must hold itself out to the public as carrying out a religious purpose, and must exercise religion consistent with, and in furtherance of, a religious purpose.
The rule defines "exercise of religion" as "any exercise of religion, whether or not compelled by, or central to, a system of religious belief" and the rule also clarifies that "the touchstone for religious exercise is sincerity, and therefore an exercise of religion must only be sincere."
The rule also states that the OFCCP can find a violation if it can prove by a preponderance of the evidence that a protected characteristic other than religion was a "but-for cause" of an employee's termination or another disciplinary measure.
Having worked for a faith-based religious organization who received three different federal grants during my time with the organization I can unequivocally state that this proposed rule is extremely helpful.
I sat through trainings in Washington, D.C. with federal grant managers who told me, point blank, that our Christian organization could not disqualify staff or potential volunteers on the basis of religion and that to do so would be in violation of the parameters of our grant. We understood that the grant could not fund activities that were expressly religious in nature (like worship services, bible studies – our grants funded our mentoring programs for children of prisoners and juvenile offenders), but in order to remain distinct it was necessary that our staff and volunteers be like-minded.
This rule change is needed and kudos to the Trump administration for proposing it.
Ernst: Let's Ensure Iowa Has the Resources to Continue to Feed and Fuel the World
U.S. Senator Joni Ernst, R-Iowa, spoke about her efforts to support Iowa's agriculture community, protect workers, and maintain a strong food supply chain.
Election Central
Kiss Your Freedom Goodbye
Phil Bair: If Democrats regain hegemony in the upcoming elections, and if they get their way on healthcare alone, we can kiss our freedom goodbye.
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Should House Democrats impeach President Donald Trump, a Senate impeachment trial could negatively impact U.S. Senators running for president.
Vander Hart: Moving the Goalposts on COVID-19
Shane Vander Hart: "Flatten the curve" has become "crush the curve" and "slow the spread" has become "stop the spread." And the COVID-19 goalposts are moved.


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text:
\section{Introduction}

One of the oldest open problems today, known as the Goldbach conjecture, is to know if every even integer greater than 2 can be expressed
as the sum of two prime numbers.
It is known since very long that the conjecture
is statistically true and it is empirically supported by calculations  for all numbers the threshold of which has gone up from
$10,000$ in 1855 \cite{Desboves} to $4\times 10^{18}$ \cite{OHP} in 2014.
In Section~\ref{sec:HaLi} we give some historical results that  support the Goldbach  conjecture.

Though the conjecture in its totality seems out of reach at the moment it generates
a lot of mathematical activity. What follows, for the most part {\it expository}, concerns only a few of these aspects while  many are obviously omitted.

Instead of studying directly the Goldbach function $g(n)=\sum_{p_{1}+p_{2}=n}1$
which counts the number of representations of an integer $n$
as the sum of two primes $p_{1}$ and $p_{2}$ and is expected to be non-zero for even $n>2$, it is convenient to treat  a smoother version using logarithms.
This is easier from the point of view of analysis and the preferred function here is  the {\it weighted } Goldbach function
\[
    G(n)=\sum_{m_{1}+m_{2}=n}\Lambda(m_{1})\Lambda(m_{2})
\]

where  $\Lambda$ denotes von Mangoldt's function
\[
   \Lambda(n)=\begin{cases}
       \log p, &\text{ if } n=p^{k} \text{ for some prime } p
   \text{ and } k\in\mathbb{N}\\ 0, &\text{ otherwise}
  \end{cases}
\]
so that  $g(n)$ can be recovered from $G(n)$ by the use of partial summation
and  the last being sufficiently large, more precisely $G(n) > C\sqrt n$, would imply the Goldbach conjecture.
As is common in analytic number theory we study the easier question  of the average order of the Goldbach functions
where the first results are at least as old as Landau's.

The dominant term in these results can be obtained easily but the oscillatory term involves infinitely many nontrivial zeros of the Riemann zeta function
 and any good asymptotic result involving upper bounds on the error term is conditional to the Riemann Hypothesis (RH). The lower bounds are
however unconditional. We indicate some such average results in Section~\ref{sec:mean}.

 Interestingly obtaining a good average order is actually equivalent to the Riemann Hypothesis.
 We include the proof  of the last statement in Section
~\ref{app:RHGo} since it has to be extracted from scattered parts in other papers.

 A variation of the classical Goldbach problem is one where the summands are
primes  in arithmetic progressions.  We present some information on the exceptional set in
Section~\ref{sec:ExcSetsAP}
and a proof in Section
~\ref{app:lem1}.  In this context average orders with good error terms are  necessarily conditional to the appropriate
Generalised Riemann Hypothesis (GRH). Equivalences with such hypotheses till
now exist only in special cases (Theorem \ref{th:BHMS2})
and seem difficult in general because of the possible Siegel zeros. Some of this is explained in Section~\ref{sec:SZe}.

\section{Goldbach Conjecture is often true}
\label{sec:HaLi}

\subsection{Hardy--Littlewood conjecture}
\label{sec:HL}

Hardy and Littlewood \cite{HL} pursued the ideas of Hardy--Ramanujan\cite {HR}
and expected an asymptotic
formula  to hold for $G(n)$, which is,
\begin{conj}[Hardy--Littlewood]
\label{conj:HL}
  The approximation $G(n)\sim J(n)$ holds for even $n$ with
  \[
  J(n):= n C_{2} \prod_{\substack{p\mid n \\ p>2}} \frac{p-1}{p-2}
  \]
and $C_{2}:=2\prod_{p>2}\Big(1-\frac{1}{(p-1)^{2}}\Big)$.
\end{conj}

The above $C_2$  is known as the twin prime constant and is approximately 1.32.
Our interest is largely in the error term
\[
  F(n):= G(n)-J(n)
\]
of the above conjecture,
where $J(n):=0$ for odd $n$.
What can be said about $F(n)$? How can we estimate it to show that it
is indeed small? Where are the sign changes?

To tackle these questions, it is natural to consider the average
$\sum_{n\leq x} F(n)$ or the second moment $\sum_{n\leq x} F(n)^{2}$.
Motivated by Conjecture \ref{conj:HL}, we should expect
$o(x^{3})$ for the second moment and $o(x^{2})$ for the average.

\subsection{Exceptional Sets}
\label{sec:ExcAP}

Using what is now called the  Hardy--Little\-wood--Ramanujan circle method, and  assuming the GRH, Hardy and Littlewood proved \cite{HL}  the estimate
\[
   \sum_{n\leq x} F(n)^{2}\ll x^{5/2+\varepsilon}
\]
for the second moment.

From these nontrivial bounds for the second moment
we are able to deduce estimates for the number of exceptions to the representation as the sum of two primes
in the
following way.

Let $E(x)=\#\{n\le x, n\in2\mathbb{N}; \ n\neq p_{1}+p_{2}\}$
denote the size of the exceptional class depending on $x$, i.e.\ the number of even integers up to $x$ which do not satisfy the Goldbach conjecture.
For these exceptions, we have $|F(n)|\geq c n$ for some constant
$c>0$, since $J(n)\geq C_{2}n$. Therefore
\[
   x^{2}E(x) \ll
   \sum_{\substack{n\leq x\\ 2\mid n\\ n \neq p_{1}+p_{2}}} n^{2} \ll
   \sum_{n\leq x} |F(n)|^{2} \ll x^{5/2+\varepsilon},
\]
so  after a dyadic dissection $$E(x)\ll x^{1/2+\varepsilon}$$ under the GRH.

More than 60 years later  Goldston \cite{Gol}  improved the Hardy--Littlewood bound to
\begin{equation}\label{eq:GBGol}
E(x)\ll x^{1/2}\log^{3} x
\end{equation}
still under the assumption of GRH.

By now many authors have succeeded in obtaining nontrivial bounds for $\sum_{n\leq
  x}F(n)^{2}$ unconditionally and we know that the
Hardy--Littlewood conjecture is true {\it on average}.

Just after 1937 when  Vinogradov's method \cite{Vino} became available, Van der Corput, Chudakov and Estermann
\cite{vdc,chu,est} independently obtained the first unconditional estimate of the type $E(x)=o(x)$. More precisely,
they proved that
\[\sum_{n\leq x}F(n)^{2}\ll
x^{3} \log ^{-A}x\]
so that
\begin{equation}
\label{eq:cce}
E(x)\ll
x\log^{-A} x
\end{equation}
for any $A>0$.

Thus we know that the Goldbach conjecture is true {\it in a statistical sense}.

In 1975 Montgomery and Vaughan \cite{MV}, improved this by using an effective form of Gallagher's work on distribution of zeros
of $L$-functions and showed
that there exists a positive effectively computable constant $\delta>0$ such that,
for all large $x>x_0(\delta)$, $$E(x)\le x^{1-\delta}.$$

The best published proof as of now is
with $\delta=0.121$, i.e.\
\begin{equation}
\label{eq:lu}
E(x)\ll x^{0.879}
\end{equation}
by Lu \cite{Lu} who obtained this by a variation of the circle method.
In the very recent preprint \cite{pin} of Pintz  the bound
\begin{equation}
  \label{eq:pin}
  E(x)\ll x^{0.72}
\end{equation}
is achieved, i.e.\ $\delta=0.28$.

Further information can be found in the survey article \cite{VauOver}
of Vaughan.

\subsection{Average Orders}\label{sec:mean}
As early as 1900 an asymptote for the average order of $g(n)$ was known
due to Landau \cite{Land} who showed that

\[
\sum_{n\le x} g(n)\sim \frac{1}{2}\frac{x^2}{\log^{2} x}.
\]
His result agreed with the conjecture that $g(n)$ should be
approximated by $\frac{J(n)}{\log^2 n}$.

After almost a century  Fujii \cite{fuj,fuj2} studied the  oscillating term. He first obtained, assuming the RH, that
\[\sum_{n\leq x}F(n)=O(x^{3/2})
\]
using the work of Gallagher of 1989.

Fujii could then extract  an error term smaller than the main oscillating term
  \cite{fuj2}, i.e.\
\begin{theorem}[Fujii's theorem]
 Assume RH. Then, for $x$ sufficiently large, we have
   \[
       \sum_{n\leq x}F(n) =-4x^{3/2}\Re\Big(\sum_{\gamma>0}
       \frac{x^{i\gamma}}{(1/2+i\gamma)(3/2+i\gamma)}\Big)
      + O( (x\log x)^{4/3})
   \]
   where $\gamma$ denotes the imaginary parts of the zeros of the Riemann zeta function.
\end{theorem}

In 2007-8 Granville \cite{Gra} also studied the average
$\sum_{n\leq x} F(n)$ and obtained the same error term.

It was conjectured by Egami and Matsumoto that the error term would be $O(x^{1+\epsilon})$ for $\epsilon$ positive
and this  was reached by Bhowmik and Schlage-Puchta \cite{B_SP} using the distribution of primes in short intervals to estimate exponential sums
close to 0. More precisely, under the assumption of the RH,  the asymptotic result  therein can be stated in the form
$$
\sum_{n\le x}G(n)=\frac{1}{2}x^2 + H(x) +O(x\log^5 x)
$$
where $H(x)=-2\sum_{\rho}\frac{x^{\rho +1}}{\rho(\rho +1)}$ is the oscillating term involving $\rho$, the non-trivial zeros of the Riemann zeta
function.

Languasco and Zaccagnini \cite{LaZa} used the circle method to improve the power of the logarithm in the error term from $5$ to $3$.

Goldston and Yang \cite{GY} could also reach $O(x\log^{3} x)$ in 2017 using the method of
\cite{B_SP}.

\subsection{Equivalence with RH}

While it is now clear that error terms for the average of $G(n)$ depends on the zeros of the Riemann zeta function  $\zeta(s)$ and hence
is meaningful only assuming
a suitable hypothesis, it is interesting to see whether we can obtain information on the zeros of $\zeta(s)$ if we have an asymptotic expansion.
This was investigated by Granville in 2007 who  stated  that (Theorem 1A  in \cite{Gra})
\begin{theorem}
\label{th:AG1A}
The  RH is equivalent to the estimate $$\sum_{n\leq x} F(n) \ll x^{3/2+o(1)}.$$
\end{theorem}

 The sketch of proof in \cite{Gra} is not sufficient to obtain the RH from the above asymptote.
 However in the recent paper  by Bhowmik, Halup\-czok, Matsumoto and Suzuki \cite{BHMS}
we were able to reconstruct part of the proof
of this theorem and it was completed with Ruzsa in \cite{BhRu}. Some details are given in Section~\ref{app:RHGo}.

\subsection {Omega-results}

The lower bounds arise while showing that there exist $n$ for which
$G(n)$ is large. Since the two main terms in the asymptotic expansion of
the average are continuous this already contributes to the error term.
It is to be noted that the lower bounds obtained are uncondtional unlike the upper ones.

The first observation seems to be that of Prachar back in 1954 \cite{Pra}
when he proved that there are infinitely many integers $n$ such that
$$ g(n)> C\frac{n}{\log^2 n} \log\log n.$$
His study was on the lines of  Erd\H os earlier for the number of solutions of $n$ as the sum of 2 prime squares with combinatorial arguments for the number of primes in residue classes modulo an integer with many prime factors.

Half a century later  Giordano \cite{Gio}  studied the irregularity of $g(n)$
depending on whether or not $n$ is divisible by many small primes by using
the Prime Number Theorem for arithmetic progression.

Having overlooked these earlier results the authors of \cite{B_SP} again showed that the error term in $\sum_{n\le x}G(n)$ is $ \Omega (x \log \log x)$
by considering the exceptional bounded modulus for which a Siegel zero might exist and using Gallagher's density estimates.

Another way to find an omega term is to study the natural boundary of the generating function $\sum_n\frac{G(n)}{n^s}$ as was mentioned in \cite{B_SP11}.

\section{Goldbach representations
in arithmetic progressions}

As a restricted form  of the original problem one considers the possibility of representing every even number as the sum
of two primes in a given residue class.
Here again the conjecture is known to be almost always true. The exceptional set
$$E(x;q,a,b)=\#\{n\le x, n\in2\mathbb{N}; \ n\neq p_{1}+p_{2}; \ p_1\equiv a(q), p_2\equiv b(q)\}$$
is shown, for example in \cite{LZ}, to satisfy, for an effectively  computable positve $\delta$
$$ E(x;q,a,b)\ll \frac{x^{1-\delta}}{\phi(q)}$$
for all $q \le x^{\delta}$.

\subsection{Exceptional Sets}
\label{sec:ExcSetsAP}

We can apply the results of the estimation of the exceptional set of the unrestricted Goldbach representation
to obtain similar results for an arithmetic progression
with residue $h$ mod $q$.
 Similar to $E(x;q,a,b)$, we define
\[
  E_{h,q}(x):=\#\{n\le x, n\in2\mathbb{N}; \ n\neq p_{1}+p_{2}; \ n\equiv h(q)\}.
\]
We show that  elementary methods already suffice to deduce
 nontrivial
estimates for the number of exceptions in an arithmetic progression
from available nontrivial estimates for $E(x)$.

Thus a very simple deduction from \eqref{eq:cce}  gives
the statement that for \emph{almost all residues} $h$ mod $q$ we have
\begin{equation}\label{trivial_hq}
    E_{h,q}(x)\ll \frac{x}{q \log ^{C} x}
\end{equation}
arising from the fact that $\sum_{0\leq h<q} E_{h,q}(x)=E(x)$ so that if
\[
   H_{q}:=\{0\leq h<q;\ E_{h,q}(x)>xq^{-1}\log ^{-C} x\}
\]
denotes the set of exceptional residues mod $q$, we have
\[
    \# H_{q} \frac{x}{q \log ^{C} x} < \sum_{h\in H_{q}}
    E_{h,q}(x) \leq E(x) \ll x(\log ^{-2C} x),
\]
and hence $\# H_{q}\ll q \log ^{-C} x $.

We remark that  the estimate (\ref{trivial_hq}) above,
for all $h$ mod $q$, is already nontrivial for modulii $q$ with
$q\gg \log ^{C} x$ for any $C>0$ which is equivalent to the condition
\[
   \frac{x}{q\log ^{C} x} \ll \frac{x}{\log ^{2C}x}.\]

Let us now introduce a handy notation. If $B\subseteq
\mathbb{N}$, we write

$B(x):=\#\{n\leq x;\ n\in B\}$ and $B_{h,q}(x):=\#\{n\leq x;\ n\in
B,\ n\equiv~ h\;(q)\}$.

\begin{lemma}[Number of exceptions in progressions]
\label{lem:l}
  Let $B\subseteq \mathbb{N}$ be such that
  $B(x) \ll x \log ^{-A} x$ for any
  $A>0$. Then for all $C>0$ and for almost all $q\leq x^{1/2}$ we have
  $B_{h,q}(x) \ll xq^{-1}\log ^{-C}x$,
uniformly in all residues $h$ mod $q$.
\end{lemma}

A proof is given in
Section~\ref{app:lem1}.

 From this Lemma and \eqref{eq:cce} we know that for any $C>0$, and for almost all $q\leq x^{1/2}$
we have
 $E_{h,q}(x)\ll xq^{-1}\log ^{-C}x$ unconditionally
for all $h$ mod $q$.

It follows immediately that for \emph  {almost all} modulii $q$ with
\\linebreak
$x^{1/2}\log ^{D}x\ll q\ll x^{1/2}(\log x)^{D}$,
we have
$$E_{h,q}(x)\ll x^{1/2}\log ^{C}x$$
for
all $h$ mod $q$, a  bound  known \emph {for all} modulii
under the assumption of the Riemann hypothesis.

Assuming RH, so that \eqref{eq:GBGol} is available,
we conclude  as in the proof of Lemma \ref{lem:l},
 that for almost all $q$ with
  $x^{1/2}\ll q\leq x^{1/2}$
  we have $$E_{h,q}(x)\ll x^{7/8+\varepsilon}/q\ll x^{3/8+\varepsilon}$$ for all $h$
  mod $q$.
(Since by \eqref{eq:ea2}, for the number $\#\mathcal{M}_{Q}$ of
exceptions in question,
$\#\mathcal{M}_{Q}\cdot x^{7/8+\varepsilon}/Q\ll x^{3/4}E(x)^{1/4}(\log
x)\ll x^{7/8}\log^{2} x$, so $\#\mathcal{M}_{Q}=o(Q)$.)

Working with the best published unconditional bound \eqref{eq:lu}
 we can likewise deduce that
$$E_{h,q}(x)\ll x^{0.97+\varepsilon}/q\ll x^{0.47+\varepsilon}$$ for all $h$
  mod $q$ and almost all  $q$ with
  $x^{1/2}\ll q\leq x^{1/2}$.

If we were working with the latest available bound \eqref{eq:pin} instead,
we would deduce
\[ E_{h,q}(x)\ll x^{0.93+\varepsilon}/q\ll x^{0.43+\varepsilon}
\]
for all $h$ mod $q$ and almost all  $q$ with $x^{1/2}\ll q\leq x^{1/2}$.

\subsection{Mean Value}

In \cite{Gra}, Granville studied  the mean value of the Goldbach
representation number $G(n)$ in an arithmetic progression,
that is the sum
\begin{equation}
\label{eq:GBap}
   \sum_{\substack{n\leq x\\n\equiv c\:(q)}} G(n),
\end{equation}
in particular for the fixed residues $c=2$ and $c=0$ and
stated
 the estimate
$\sum_{n\leq x, n\equiv 2\:(q)} F(n)\ll x^{3/2+o(1)}$.
We introduced
 the technically useful sum \cite{BHMS}
\[
S(x;q,a,b)= \sum_{n\leq x} \sum_{\substack{\ell+m=n\\\ell\equiv a,
    m\equiv b (q)}} \Lambda(\ell) \Lambda(m)
\]
a special case of which, $\sum_{a\:(q)} S(x;q,a,c-a)$ is  the above \eqref{eq:GBap},

and proved that, for $(ab,q)=1$,
\[
S(x;q,a,b)= \frac{x^2}{2\phi(q)^2} +O(x^{1+B_q})
\]
where $B_q$ depends on the non-trivial zeros of the associated Dirichlet $L$-functions.
The oscillating term was also extracted but we do not discuss it here.

\subsection{Equivalence with  GRH-DZC}

Parallel to the equivalence of the RH and the asymptotic expansion of the classical Goldbach average,
it was believed that there is an equivalence between the RH for Dirichlet
$L$-functions $L(x,\chi)$ over all characters $\chi$ mod $m$
which are odd squarefree divisors of $q$ and the estimate
$\sum_{n\leq x, n\equiv 2\:(q)} F(n)\ll x^{3/2+o(1)}$   (\cite{Gra}, Thm.1B).

In \cite{BHMS} the same question was studied and the above claim  was recovered
though partially, i.e.\
only in the case $b=a$, and additionally under the assumption of one more conjecture
on the non-trivial zeros of $L$-functions, called the Distinct Zero Conjecture
(DZC), more precisely,
\begin{quote}
\textit{For any $q\geq 1$, any two distinct Dirichlet $L$-functions
associated with characters of modulus $q$
do not have a common non-trivial zero, except for a possible
multiple zero at $s=1/2$.}
\end{quote}

The equivalence obtained till now does not cover all residues $a$ and $b$.

\begin{theorem}[{from \cite[Thm.1]{BHMS}}]
\label{th:BHMS1}
Let {\upshape DZC} be true, $q$ be odd and $(a,q)=~1$.
Then, for any $\varepsilon>0$, the asymptotic formula
\begin{align}
\label{MainTheorem-1-formula}
S(x;q,a,a)=\frac{x^2}{2\varphi(q)^2}+O_{q,\varepsilon}(x^{3/2+\varepsilon})
\end{align}
is equivalent to the GRH for the functions $L(s,\chi)$ with any character $\chi$ mod $q$.
\end{theorem}
Continuing in the direction of possible equivalences
a bit more can be said using the notation
$B_{\chi}=\sup\{\Re \rho_{\chi}\}$ and $B_q=\sup\{B_{\chi}\mid\chi\:(q)\}$
where $\rho_{\chi}$ are the non-trivial zeros of $L(s, \chi)$.

\begin{theorem}[{from \cite[Thm.3]{BHMS}}]
\label{th:BHMS2}
  Let $q,c$ be integers with $(2,q)\mid c$.

Assuming the GRH, we have
\begin{equation}
\label{thmsiegel_asymp}
\sum_{\substack{n\leq x\\n\equiv c\,(q)}}F(n)\ll x^{3/2}.
\end{equation}

On the other hand if we assume that
\begin{equation}
\label{eq:thmsiegel}
\sum_{\substack{n\leq x\\n\equiv c\:(q)}}F(n)\ll_{q,\varepsilon}
x^{3/2+\varepsilon}
\end{equation}
holds for any $\varepsilon>0$ and
that there exists a zero $\rho_0$ of $\prod_{\chi\:(q)}L(s,\chi)$ such that
\begin{enumerate}[(a)]
\item $B_q=\Re\rho_0$
\item $\rho_0$ belongs to a unique character $\chi_1\ (q)$
\item the conductor $q^\ast$ of $\chi_1\ (q)$ is squarefree and
  satisfies $(c,q^\ast)=1$,
\end{enumerate}
then $B_q=\Re \rho_0\le 1/2$.
\end{theorem}
Thus the GRH can be deduced only under several additional assumptions (a),(b) and (c),
from the asymptotic expansion \eqref{eq:thmsiegel}.

It is worth mentioning that  from  \eqref{thmsiegel_asymp} above, it
is immediate that \linebreak
$\sum_{n\leq x}F(n)\ll x^{3/2}$ when $q=1$ which in turn implies that under the RH
 we get the bound $E(x)\ll x^{1/2}\log x$. The last is an  improvement on Goldston's result \eqref{eq:GBGol}.

\subsection{Bombieri--Vinogradov Theorem and Siegel Zeros}
\label{sec:SZe}

Under the assumption that the
GRH can be deduced
for all $L(s,\chi)$ for any $\chi$ mod $q$ from asymptotic expansions
we would have  a very short
proof of Bombieri--Vinogadov's theorem. Let us dwell on this possibility.

Let $D_{q}(x):=\sum_{\substack{n\leq x \\ n\equiv 2(q)}} (G(n)-J(n))$
and $\Delta_{q}(x):= \max_{a(q)}^{*}
|\psi(x;q,a)-\frac{x}{\varphi(q)}|$,
where trivially $\Delta_{q}(x)\ll xQ^{-1}\log ^{2}x$.
Let
\[
   E:=\{q\in\mathbb{N};\ Q<q<2Q,\ |D_{q}(x)|\geq x^{3/2+\delta} \text{ for some
   }\delta>0 \},
\]
so that $q\not\in E \implies D_{q}(x)\ll x^{3/2+o(1)}$. From our assumption
this would imply GRH for $L(s,\chi)$ with any $\chi$ mod $q$.
Then $\Delta_{q}(x)\ll x^{1/2}\log ^{2}x$.

Hence
\[
   \# E\, x^{3/2+\delta} \leq \sum_{q\in E} |D_{q}(x)|,
\]
and by Lemma \ref{lem:bv} below we would have
\[
\sum_{q\in E} |D_{q}(x)| \leq \Big( \sum_{n\leq x} |G(n)-J(n)|^{2}\Big)^{1/2}
x^{1/2} \log ^{3/2}x\ll x^{2} \log ^{7/2} x
\]
by trivially estimating the last sum as $x^{3}\log ^{4}x$.
From this, we could deduce that $\#E \ll x^{1/2-\delta}\log ^{4} x $.

For the left hand side of the Bombieri--Vinogradov's theorem this yields
\begin{align*}
  \sum_{Q<q\leq 2Q} \Delta_{q}(x)
  &=\sum_{q\in E} \Delta_{q}(x) + \sum_{q\not\in E} \Delta_{q}(x) \\
  &\ll x^{1/2-\delta} xQ^{-1}\log ^{6}x + Qx^{1/2}\log ^{2}x \ll
  x\log ^{-A}x
\end{align*}
if we assume that $x^{1/2-\delta}\log ^{6+A}x\ll Q\ll x^{1/2}\log
^{-2-A}x$.

The elementary lemma used just above is  :
\begin{lemma}
  \label{lem:bv}
  For any sequence $(v_{n})_{n\in \mathbb{N}}$ of complex numbers
and any integer $a$ with $a<x$ we have
\[
   \sum_{Q<q\leq 2Q} \Big|\sum_{\substack{n\leq x\\n\equiv a(q)}} v_{n}\Big|
   \leq \Big( \sum_{n\leq x} |v_{n}|^{2}\Big)^{1/2} x^{1/2} \log ^{3/2}x.
\]
\end{lemma}

\begin{proof}
  The left hand side is equal to the sum $\sum_{Q<q\leq 2Q} |\langle
  v,\varphi_{q}\rangle|$ with $\varphi_{q}(n)=1$ if $n\equiv a$ mod
  $q$ and $\varphi_{q}(n)=0$ otherwise. By Halasz--Montgomery's
  inequality, this is
\[
   \leq \Big(\sum_{n\leq x} |v_{n}|^{2}\Big)^{1/2} \Big(\sum_{q_{1},q_{2}} \langle
   \varphi_{q_{1}},\varphi_{q_{2}}\rangle \Big)^{1/2}
\]
with
\[
\langle\varphi_{q_{1}},\varphi_{q_{2}}\rangle = \sum_{n\leq x}
\sum_{q_{1}\mid n-a} \sum_{q_{2}\mid n-a} 1 = \sum_{s, 0<x-a\leq s\leq
2x-a } \sum_{q_{1}\mid s} \sum_{q_{2}\mid s} 1 \ll x \log ^{3}x.
\]
\end{proof}

Concerning the Siegel zeros,  Fei in \cite{fei} has studied a similar question.
Assuming
a certain version of
\emph{weak} Goldbach conjecture, namely
if for all even $n>2$,
\[
  G(n)\geq \frac{\delta n}{\log^{2} n},
\]
then the possible Siegel zero $\beta$ for $\chi$ mod $q$, where
$q$ is a prime, $q\equiv 3$ mod $4$ and satisfies
\[
  \beta\leq 1-\frac{c}{\log^{2} q}
\]
for some constant $c>0$.
Thus here we get a repulsion of the Siegel zero from the line $\Re s=1$ while
assuming \eqref{thmsiegel_asymp}
 on average
we deduce $B_{q}\leq 1/2$ under (a), (b) and (c), which is a
repulsion of all the nontrivial zeros of $L$-functions mod $q$
from the line $\Re s=1$.

\newcommand{\,\textup{d}u}{\,\textup{d}u}
\section
Proof of equivalence of RH and
Goldbach average }
\label{app:RHGo}

In this section, we give an overview of the proof of Theorem
\ref{th:AG1A}, that the Riemann Hypothesis  is equivalent to the estimate
\begin{equation}
\label{eq:f}
\sum_{n\leq x} (G(n)-J(n)) \ll_{\varepsilon} x^{3/2+\varepsilon}
\end{equation}
for any $\varepsilon>0$.

Since $\sum_{n\leq x}J(n)-x^{2}/2\ll x \log x$,
we can write  \eqref{eq:f} equivalently  as
\begin{equation}
  \label{eq:ff}
  \sum_{n\leq x} G(n) =\frac{x^{2}}{2}+O(x^{3/2+\varepsilon})
\end{equation}
for any $\varepsilon>0$.

We concentrate on the proof of the deduction of the RH from
 \eqref{eq:f}
that is to obtain $B=1/2$ for
$B:=\sup \{\Re \rho;\ \zeta(\rho)=0\}$.

Step 1. Let
$S(x):=\sum_{n\leq x} G(n)$ be the summatory function of $G(n)$.
A key issue is the proof of the asymptotic formula
\[
   S(x)=x^{2}/2+\sum_{\rho} r(\rho)
   \frac{x^{\rho+1}}{\rho+1} + E(x)
\]
with $E(x)\ll x^{2B}\log ^{5} 2x$
and $r(\rho):=-2/\rho$
( Theorem 2 in \cite{BHMS}). This involves a careful
transformation of the problem to an exponential sum setting where
Gallagher's lemma can be used, confer \cite[Lemma 9]{BHMS}.

Step 2.
We define the Goldbach generating Dirichlet series as
\[
F(s)=\sum_{n=1}^{\infty}\frac{G(n)}{n^s}.
\]
This can be computed by using $S(x)$ in the integral
\[
   F(s)=s\int_{1}^{\infty} S(u)u^{-s-1} \,\textup{d}u,
\]
where inserting the formula from Step 1 yields
\begin{align}
  F(s)
&=\frac{s}{2(s-2)}+\sum_{\rho}\frac{r(\rho)s}{(\rho+1)(s-\rho-1)}
+s\int_1^\infty E(u)u^{-s-1} \,\textup{d}u \notag\\
&=\frac{1}{s-2}+\sum_{\rho}\frac{r(\rho)}{s-\rho-1}
+s\int_1^\infty E(u)u^{-s-1}\,\textup{d}u+C_1, \label{eq:fs}
\end{align}
with
\[
C_{1}=\frac{1}{2}+\sum_{\rho}  \frac{r(\rho)}{\rho+1}
\]
and $r(\rho)=-2/\rho$.

From the above we can read off that for $\sigma>2$, the function $F(s)$
converges absolutely and is {\it analytic}.

Moreover $F(s)$ can be {\em continued meromorphically}
to the half plane $\sigma>2B$ since $E(u)\ll u^{2B}\log^{5} (2u)$.

Step 3. Assume $B<1$. Then we have
\begin{equation}
\label{eq:FB}
 1+B=\inf\{\sigma_0\ge\frac{3}{2} \mid
F(s)-\frac{1}{s-2}\text{ is analytic on }
  \sigma>\sigma_0\}.
\end{equation}
 Step 2 shows that the infimum on the right is at most $2B\leq B+1$,

For the inequality in the other sense we observe that this is trivially true for $B=1/2$,
so we may assume that $1/2<B<1$.

Now $\max(2B,3/2)<1+B$ being a strict inequality, there exists a
real number $\varepsilon>0$ such that $\max(2B,3/2)<1+B-\varepsilon$ holds true.
Then by the definition of $B$, there exists a zero $\rho$ such that
$1/2<B-\varepsilon<\Re\rho$.

From the formula for $F(s)$ from Step 2, the function
 has a pole at $\rho+1$ with
residue $r(\rho)=-2/\rho\neq 0$ in the half plane
$\sigma>1+B-\varepsilon>3/2$, and we conclude that
\[
 1+B-\varepsilon\leq\inf\{\sigma_0\ge\frac{3}{2}\mid
F(s)-\frac{1}{s-2}\text{ is analytic on }
  \sigma>\sigma_0\}.
\]
Letting $\varepsilon\to 0$, we obtain  the desired goal.

Step 4.
Now let $D(x)=\sum_{n\leq x} G(n) -\frac{x^{2}}{2}$,
and we have $D(x)\ll x^{3/2+\varepsilon}$ from \eqref{eq:ff}.
Hence, as in the proof of \eqref{eq:fs},
\[
F(s)-\frac{1}{s-2}
=
s\int_1^{\infty}D(u)u^{-s-1}\,\textup{d}u+\frac{1}{2}
\]
for $\sigma>2$,
where the right-hand side gives an analytic function on $\sigma>3/2$ since
$D(u)\ll u^{3/2+\varepsilon}$ from \eqref{eq:ff}.

Therefore, by \eqref{eq:FB} from Step 3,
we conclude that $B\leq 1/2$ provided that $B<1$, hence RH.

Step 5.
We now need to exclude the possibility that  \eqref{eq:ff} could imply $B=1$
(see \cite{BhRu}).
For this, let $|z|<1$ and consider the power series
\[
   f(z)=\sum_{n\geq 1} \Lambda(n)z^{n} \text{ and }
   f^{2}(z)=\sum_{n\geq 1} G(n)z^{n},
\]
so that
\[
  \frac{1}{1-z}f^{2}(z)=\sum_{n\geq 1} S(n)z^{n},
\]
again with the summatory function
$S(x)=\sum_{m\leq x}G(m)$ of the Goldbach function $G(m)$.
Then
\[
    \frac{1}{1-z}f^{2}(z)=\frac{1}{(1-z)^{3}}+O(N^{5/2+\varepsilon})
\]
on the circle $|z|=e^{-1/N}$, which can be reformulated as
\[
    f^{2}(z)=\frac{1}{(1-z)^{2}}+O(|1-z|N^{5/2+\varepsilon}).
\]

This yields an asymptotic formula
on the major arc $|1-z|\leq cN^{-C/3}$.
Taking the complex {\emph square root} yields
\[
   f(z)= \pm \frac{1}{1-z}+O(|1-z|^{2}N^{5/2+\varepsilon}).
\]
Due to continuity and non-negativity of the coefficients of $f(z)$,
we have the plus sign throughout the whole major arc.

\newcommand{\,\textup{d}z}{\,\textup{d}z}
Now by Cauchy's integral formula, we obtain
\[
\psi(N)
=
\frac{1}{2\pi i} \int_{|z|=R} f(z) K(z)\,\textup{d}z,\quad
N
=
\int_{|z|=R}\frac{1}{1-z}K(z)\,\textup{d}z
\]
for the kernel
\[
K(z)=z^{-N-1}+z^{-N}+\dots+z^{-2}=z^{-N-1}\frac{1-z^{N}}{1-z}.
\]

The contribution of $f(z)$
to this integral is $O(N^{5/6})$ on the
major arc, and only
$O(N^{11/12+\varepsilon})$
for the minor arc (which needs a little care to
prove).

Comparing this with the explicit formula for $\psi(N)$, we conclude
that $B<11/12<1$.

\section
Proof of Lemma
on the number of exceptions in progressions}
\label{app:lem1}
For a positive integer $N$, for $a_{1},\dots,a_{N}\in \mathbb{C}$
and a residue $h$ mod $q$ denote
\[
   Z:=\sum_{n\leq N}a_{n} \text{ and } Z(q,h):=\sum_{\substack{n\leq
       N\\n\equiv h\ (q)}}a_{n}.
\]

We start with the proof of the following
\begin{theorem}
  \label{t2}
 For any real $H>0$ and $Q>1$ we have
   \begin{align*}
      \sum_{Q<m\leq 2Q} m &\max_{h\text{ \emph{mod} }m}
       |Z(m,h)|^{2} \\ &\leq (N^{2}+Q^{2})
      \frac{\log Q}{H} \max_{n\leq N} |a_{n}|^{2}
      + (N+Q^{2})H\log Q \sum_{n\leq N}|a_{n}|^{2}.
   \end{align*}
\end{theorem}

\begin{proof}
We write $m\sim Q$ for $Q<m\leq 2Q$.
Split the left hand side of the theorem into
$E_{1}+E_{2}$ with
\begin{equation*}
  E_{1}:= \sum_{\substack{m\sim Q\\\tau(m)>H}}
    m \max_{h \text{ mod } m}  |Z(m,h)|^{2}
\end{equation*}
and
\begin{equation*}
  E_{2}:= \sum_{\substack{m\sim Q\\\tau(m)\leq H}}
    m \max_{h \text{ mod } m}  |Z(m,h)|^{2},
\end{equation*}
where $\tau(m)$ denotes the number of divisors of $m$.

Consider first $E_{1}$. Let
\begin{equation*}
  A:=\#\{ m\sim Q; \;\tau(m) > H \},
\end{equation*}
then
\begin{equation*}
   A H <
  \sum_{\substack{m\sim Q \\ \tau(m)> H}} \tau(m)
  \leq\sum_{m\leq 2Q} \tau(m) \ll  Q\log Q,
\end{equation*}
so
\begin{equation*}
  A\ll \frac{Q\log Q}{H}.
\end{equation*}
Since $Z(m,h)\ll(\frac{N}{m}+1)\max_{n\leq N} |a_{n}|$ we get
\begin{align*}
  E_{1}&\ll \sum_{\substack{m \sim Q \\ \tau(m)>H }}
      m \max_{h} |Z(m,h)|^{2} \ll
      \sum_{\substack{m \sim Q \\ \tau(m)>H }} m\biggl(\frac{N^{2}}{m^{2}}
      +1\biggr) \max_{n\leq N} |a_{n}|^{2}  \\ &\ll
      A \biggl(\frac{N^{2}}{Q} +Q\biggr)\max_{n\leq N} |a_{n}|^{2}
      \ll \biggl(\frac{N^{2}}{H} +\frac{Q^{2}}{H}\biggr) \log Q
      \max_{n\leq N} |a_{n}|^{2}.
\end{align*}
This is the first summand on the right hand side of Theorem \ref{t2}.

\vspace{\baselineskip}
Now we look at $E_{2}$. From Theorem \ref{t1} below we have
\[
  E_{2}=\sum_{\substack{m\sim Q\\ \tau(m)\leq H}} m
   \max_{0< h\leq m} |Z(m,h)|^{2}
   \leq \sum_{\substack{d\leq 2Q}} M'_{d} \sum_{\substack{ 0<b\leq d \\
     (b,d)=1}} \Big| T\Big( \frac{b}{d} \Big)\Big|^{2}.
\]
Now we estimate
\[
   M'_{d}= \sum_{\substack{m\sim Q,d|m\\\tau(m)\leq H}}
   \frac{\tau(m)d}{m}\ll H\log Q
\]
and an application of the {\em large sieve inequality} yields
\[
   E_{2}\ll H \log Q\; (N+Q^{2})\sum_{n\leq N}|a_{n}|^{2},
\]
which is the second term on the right hand side of Theorem
\ref{t2}.
\end{proof}

We use the following result.
\begin{theorem}
\label{t1}
  We have the estimates
  \[
      \sum_{m\in\mathcal{M}} m \max_{h \text{ \emph{mod} }m} \Big|
      Z(m,h)-\frac{Z}{m}\Big|^{2} \leq \sum_{d=2}^{\infty} M'_{d}
      \sum_{\substack{ 0<b<d \\ (b,d)=1}}
      \Big|T\Big(\frac{b}{d}\Big)\Big|^{2}
  \] and
  \[
     \sum_{m\in\mathcal{M}} m \max_{h \text{ \emph{mod} }m}
      |Z(m,h)|^{2} \leq \sum_{d=1}^{\infty} M'_{d}
      \sum_{\substack{ 0<b\leq d \\ (b,d)=1}}
      \Big|T\Big(\frac{b}{d}\Big)\Big|^{2}
  \]
  with
  \[
     M'_{d} := \sum_{t, td\in\mathcal{M}} \frac{\tau(dt)}{t}.
  \]
\end{theorem}
Here $\tau$ denotes the divisor function.
Note that $M'_{d}\ll \tau(d)\log^{2}Q$ if
$\mathcal{M}\subseteq\{1,\dots,\lfloor Q\rfloor\}$ for a real number
$Q>1$.

\begin{proof}
   Let
\[
   f_{h}(m) := \sum_{d|m} \mu(d) \frac{m}{d} Z\Bigl(\frac{m}{d},h\Bigr),
\]
then we have by M\"obius inversion
\begin{align*}
  &\sum_{m} m \max_{h} |Z(m,h)-Z/m|^{2} =  \sum_{m} \frac{1}{m}
  \max_{h} |mZ(m,h)-Z|^{2} \\&= \sum_{m} \frac{1}{m} \max_{h}
   \Big|\sum_{d|m} f_{h}(d)-Z\Big|^{2} = \sum_{m} \frac{1}{m}
   \max_{h}\Big|\sum_{\substack{ d|m\\d\neq 1}} f_{h}(d)\Big|^{2}\\
   &\leq \sum_{m} \frac{\tau(m)}{m} \sum_{\substack{ d|m\\d\neq 1}}
   \max_{h}|f_{h}(d)|^{2}.
\end{align*}
Now we note that $f_{h}(d)$ is $d$-periodic in $h$ for $d|m$,
since  $Z(t,h+d)=Z(t,h)$ for $t|d$, so
\begin{multline*}
  f_{h+dl}(d)=\sum_{t|d}\mu(t) \frac{d}{t}
  Z\Bigl(\frac{d}{t},h+dl\Bigr) = \sum_{t|d}\mu(t) \frac{d}{t}
  Z\Bigl(\frac{d}{t},h\Bigr) = f_{h}(d)\\ \text{ for all } l\in\mathbb{Z},
\end{multline*}
therefore the maximum remains the same if taken only over $h$ with
$0< h\leq d$. We estimate this maximum by $\sum_{0< h\leq d}$,
therefore an upper estimate for $\max_{0< h\leq q}|f_{h}(d)|^{2}$ is
\[
\sum_{h=1}^{d} |f_{h}(d)|^{2} = d\sum_{\substack{0\leq b \leq d\\(b,d)=1}}
\Big|T\Big(\frac{b}{d}\Big)\Big|^{2}
\]
by Montgomery's formula \cite{mf}, namely
\begin{equation*}
q\sum_{h=1}^{q}\Big| \sum_{d|q} \frac{\mu(d)}{d}
   Z\Big(\frac{q}{d},h\Big) \Big| ^{2}
   = \sum_{\substack{ 1\leq a\leq q \\ (a,q)=1}}
   \Big|T\Big(\frac{a}{q}\Big)\Big|^{2}
\end{equation*}
for the exponential sum
\[
   T(\alpha):=\sum_{n\leq N} a_{n} e(\alpha n).
\]

So we get
\[
\sum_{m} m \max_{h} |Z(m,h)-Z/m|^{2} \leq \sum_{d=2}^{\infty} M'_{d}
\sum_{\substack{ 0<b<d \\ (b,d)=1 }} \Big|T\Big(\frac{b}{d}\Big)\Big|^{2}
\]
with
\[
  M'_{d} := \sum_{m\in\mathcal{M}, d|m} \frac{\tau(m)d}{m} =
          \sum_{t, dt\in\mathcal{M}} \frac{\tau(td)}{t}.
\]
This shows the first inequality of Theorem \ref{t1}.

The only change for the proof of the second inequality when replacing
$|Z(m,h)-Z/m|^{2}$ by $|Z(m,h)|^{2}$  is to include
the summand for $d=1$ in the sums over $d$.
\end{proof}

Now we give the proof of Lemma \ref{lem:l}.
\begin{proof}
   Let $Q\leq x^{1/2}$ and $N=\lfloor x\rfloor$.
  Then Theorem \ref{t2}, used with the indicator
   function for $B$ as sequence $(a_{n})_{n\leq N}$, shows
\begin{equation*}
    \sum_{q\sim Q} \max_{h\text{ mod }q} B_{h,q}(x) \ll
    (x^{2}H^{-1}+HxB(x))^{1/2}\log ^{1/2}Q.
\end{equation*}
The optimal choice for $H$ is $H=(x/B(x))^{1/2}$, and therefore
we have
\begin{equation}
\label{eq:ea2}
\sum_{q\sim Q} \max_{h\text{ mod }q} B_{h,q}(x)
   \ll x^{3/4}B(x)^{1/4}\log ^{1/2}Q.
\end{equation}

So if $B(x)$ is small, we expect  $B_{h,q}(x)$ to be
small too.

Consider therefore for $C>0$ the number of exceptional modulii
\[
   \mathcal{M}_{Q}:=\{q\sim Q;\ B_{h,q}(x)>\frac{x}{q\log ^{C}x}
   \text{ for any } h \text{ mod }q\}.
\]
It follows that
\begin{align*}
   &\#\mathcal{M}_{Q}\cdot\frac{x}{Q\log ^{C}x} \ll
   \sum_{q\in\mathcal{M}_{Q}} \max_{h} B_{h,q}(x) \\ &\qquad\ll x^{3/4}
   B(x)^{1/4} \log ^{1/2}Q \ll \frac{x}{\log ^{2C+1}x},
\end{align*}
so $\#\mathcal{M}_{Q}\ll \frac{Q}{\log ^{C+1}x}$.

If we split $[1,x^{1/2}]$ into $\ll \log x$ many dyadic intervals,
we get
\[
   \#\{q\leq x^{1/2};\; B_{h,q}(x)>\frac{x}{q\log ^{C}x} \text{ for
     any }h\text{ mod }q \} \ll \frac{x^{1/2}}{\log ^{C}x}.
\]
This proves the Lemma.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Not too long ago Ayia Napa was a small sleepy fishing village with a small harbour, a couple of tavernas and quiet long sandy beaches. Nearby Famagusta was the hotspot of the coastal region with its big hotels and resort mentality. Ayia Napa was fishermen and farmers, the famous Cyprus potato comes from this region, but soon after the invasion and subsequent pumping in of aid the farmers discovered that their coastal land was far more valuable to develop than to farm and so not so slowly hotels and tourist villages began to spring up in the area until Ayia Napa started to get an international reputation for being a clubbing paradise. The boom of tourism has transfigured it in a relatively short space of time into a throbbing night-spot and fashionable beach playground. The waters on this side of Cyprus are superior in colour, which is that lovely turquoise blue and the sand especially on Nissi beach is golden and clean. The fast food outlets and tourist shops invested in Ayia Napa, more clubbing happened, spawning more clubs which sprang up like the local potatoes, DJ's started to visit the resort and soon it was heaving. This of course made more and more farmers give up their land to the developers in exchange for a piece of the action and now only twenty years later Ayia Napa has almost met up with the neighbouring villages on both sides. Protaras to the north and Ayia Thekla/ Potamos to the south. The attractions, villas, restaurants and hotel complexes sprawl along the coast and a new highway has been built to connect to the airport in Larnaca with easy access to the capital Nicosia. Nearby at Dherinia there is a lookout point where you can see the still deserted part of Famagusta, behind the united nations lines, clearly marked by the cessation of residential areas where the fields get to grow their traditional crops in peace. It seems to be particularly appealing to young people, who are catered for quite handsomely, from lunar parks and water parks for the very young to go carts, bowls and driving ranges for the more mature.
The beaches around Ayia Napa and Protaras are famous for the beautiful blue of their water and although they are now overcrowded in some places it is still a lovely spot, especially if you like an active holiday. For those of you who like to partake of the action without being in the thick of it all the time, Protaras, Ayia Thekla or Paralimni are an ideal alternative.
In the table below we have listed some of the more traditional sights to be seen, the churches, museums and places of general interest. For a guide to what's on while you are here, including art exhibitions, concerts, folk dancing, cinema program and special tourist events see	Cyprus-art.com which used to be updated every Friday.
The mediaeval monastery dedicated to 'Our Lady of the forests' stands in the middle of the village surrounded by a high wall. Its 16th Century church is partially underground and cut into the rock. The ancient sycamore tree in front of the south gate is believed to be 600 years old. The more modern aspects of the village include hundreds of bars , restaurants , disco's and places of entertainment. It is a very popular destination. A recent development is the wedding business in Ayia Napa, a lot of young people choose to get married here.
Here is a 16th Century church dedicated to the Virgin Mary and the 15th Century church of Ayios Andronikos, with an octagonal dome and murals which can still be distinguished in the apse.
Ayios Andronikos wine is very nice indeed. Liopetri still carries on the traditional craft of basket weaving or basket making and the school for the blind in Nicosia is a place where you can still have a seat restrung or your piano tuned.
The church, in Paralimni dedicated to the Virgin Mary (Panayia), is decorated with unusual 18th Century porcelain plates. The church, parts of which belong to the 13th Century, houses a small ecclesiastical museum. Other interesting churches are also to be found in the area. Paralimni deserves more of a focus than this little note, as it is not only better for shopping than Ayia Napa, but is becoming a popular residential area. We will get to it soon.
A picturesque fishing refuge. Nearby are the remains of a Venetian watch-tower. The French poet Arthur Rimbaud worked in this area in the 1880's, read him and see if you recognise the nightlife.
The monastery in the centre of Ayia Napa is just on the edge of the square. It is a tranquil place which reeks of simpler days gone bye and is a great place to relax in the cool shade of the trees and arches. A photo opportunity indeed as you can see in the pictures to the left. At night however it has been known to heave with the young, glistening in the summer with a sea of sweaty bodies, the beer bottles glint in the moonlight and if you don't like noise, crowds or throbbing music, it is a place to avoid.
At the east entrance of this village is the partly ruined church of Ayios Mamas which was built around the 12th century and rebuilt in the 16th Century. Most of the existing surviving frescoes date from the 16th Century. About 1,5 km to the west is the 16th Century cruciform church of Panayia Chordakiotissa (mother village music??) and the church of Ayios Georgios, (Saint George) an Early Christian basilica .
The village of Dherynia has three interesting churches; there is a small domed 17th Century church named Ayios Georgios, the 15th Century Ayia Marina church, and the church dedicated to the Virgin Mary with its 17th Century icons. From a vantage lookout point above the village one may look upon the 'ghost city' of Famagusta, still under occupation by the Turkish forces. Once the most popular holiday resort in Cyprus, now with trees growing in the streets.
Protaras is the up market version of Ayia Napa. Nice beaches and interesting coves, that in itself is healthy entertainment for the children, nice hotels, lots of villas and plenty of nightlife but not till 5am. The evening winds down by 2am here.
Cyprus wedding while you are here?


Two-Days-Per-Page Planning Notebook, Gray, 6" x 9"
Works perfectly for any year. Lined planning space provides ample room for notes. HOT-Spot reminder section on each page. Three-year reference calendar, index sheet, future planning and contact pages. Double-sided pocket for loose materials.


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text:
Escaping Economic Orthodoxy in Latin America: "I think on the whole the departure has been a success."
An interview with Ha-Joon Chang on the region's alternatives to the Washington Consensus
Ha-Joon Chang is one of the world's best-known and most influential economists. A native of South Korea, he is the author of a series of international best-selling books that have inspired a new generation to rethink economics by challenging economic orthodoxy. Last year, Chang was voted one of the top 10 world thinkers by readers of the U.K.-based Prospect magazine. His books, including Bad Samaritans: The Myth of Free Trade and the Secret History of Capitalism, and 23 Things They Don't Teach You About Capitalism, have been translated into dozens of languages. He has also advised numerous U.N. agencies, institutions, corporations, NGOs, and governments on economic policy. Chang teaches economics at Cambridge University in England. NACLA's Dan Beeton recently caught up with him in Washington, D.C. when Chang was promoting his latest book, Economics: The User's Guide (Bloomsbury).
Dan Beeton: The so-called "pink tide" of left-leaning governments that have emerged in Latin America over the past 15 years have often pursued economic policies that go against the so-called Washington Consensus neoliberal policies of reduced social spending, privatization, trade liberalization and so on. How would you characterize these countries' progress? Do you think there are success stories here that other countries might want to ponder?
Ha-Joon Chang: In the 80s and 90s, all the Latin American countries followed the Washington Consensus policy and throughout those decades Latin America hardly grew at all—despite all the promises that if you follow these policies you'll grow fast and foreigners will invest. Those promises haven't been kept and a lot of countries got disillusioned, and from the late '90s to early 2000s started electing governments that departed from the Washington Consensus policies.
Now, the degree of departure varied across countries. At one extreme you have countries like Chile or Colombia, which basically still follow the neoliberal policies but did small things to modify the worst features by, for example, introducing some degree of capital control, and expanding some social policies. At the other extreme, you have Venezuela, which basically tried to find its own model, although not necessarily successfully. But in the middle we have countries ranging from Brazil to Bolivia that are trying different things, but even within the spectrum. Brazil while doing a lot of non-orthodox things, more or less maintains the orthodox version when it comes to macroeconomic policy. But in terms of social policy and industrial policy, in the last decade it has significantly departed from the Washington Consensus.
A lot more state intervention.
That's right. At the other end of the spectrum, you can talk about countries like Bolivia, Argentina, Ecuador which did more radical things to change. I think on the whole the departure has been a success. Of course you cannot attribute the growth acceleration entirely to policy changes, because a lot of it was thanks to a commodity boom due to China's super-growth. But there were clear results in other areas, like social policy, conditional cash transfers. I have some problem with those, but still, for the first time in probably five centuries, income inequality has fallen in Brazil—although only slightly—and I think you have to give credit to these policy changes. And having seen some results here, these countries are pushing in the non-orthodox direction even more. Chile's now got significant left-wing presence in politics; even Colombia's kind of moving away from the standard right-wing tradition; Argentina, thanks to its struggle with the vulture funds, has to do things differently; Ecuador and Uruguay have been slowly but surely moving further and further away from the Washington Consensus.
I think the next decade will see even more departure from these policies. But at one level I'm concerned because, basically, this Chinese commodity boom has meant that these countries could grow quite quickly without upgrading their production structures. They know they have to upgrade and start doing difficult things, but the pull is too strong. If you look at the actual structures, Ecuador is still very, very dependent on oil; Brazil's dependence on commodities has probably increased. So they have to face the fact that this commodities boom is not going to last forever. Even if it did last for a significant period, unless you use the proceeds from the commodity exports during that period to build your manufacturing base, when it finally ends after – I don't know – 20 years or whatever, you'll have no engine of growth for the future.
So they need to find new markets?
Yeah, new markets, but where? Other potential new markets are also countries dependent on primary commodities, so they need to begin to think about new things to produce. Some countries lack the capabilities to do so, but there are some Latin American countries where they have very well-trained scientists and engineers and could develop new activities. Unfortunately, there's too much ideological obsession that governments shouldn't do too much industrial policy, while the political power of commodity interest is too strong. Commodity producers really do not want industrial development in the same way that Southern plantation owners didn't want it in America in the 19th Century. So, this isn't going to be easy, but that's what they have to do.
The BRICS recently announced they would create new institutions: the Contingent Reserve Arrangement, and the New Development Bank. This has elicited much commentary and analysis, with some economists suggesting that these new institutions could offer important alternatives to the IMF and World Bank, which routinely condition lending on the adoption of neoliberal economic policies. Do you share this assessment? How significant do you think these institutions may be?
Well [that] depends on how much money's there. In the end, money's what makes those institutions like the World Bank and IMF powerful. Of course, it's not just that they're owed money; they're even more powerful than the money they have because the powerful countries that control these organizations will add money only if the IMF gives its seal of approval. So it's not just money—it's also soft power.
A BRICS bank may have an equal amount of money as the World Bank, but if the other potential public and private funders do not agree with its assessment, its power will be weaker than that of the World Bank. In general, yes, having more alternatives gives weaker countries more bargaining power. In the '80s and '90s, countries in Latin America and Africa had to listen to the IMF and World Bank because they were the only banks in town, so to speak. Now there's another bank called "China"; there might be another bank called "BRICS." If you have alternatives, you can play them against each other and get better terms. I think this way at least some African countries have already benefitted from the presence of China—if not the forthcoming BRICS bank—because the Chinese, when they lend money, they don't put all those conditions that the World Bank and the IMF place on their loans.
So, yes, I think on the whole it's a good thing. How important they will be depends on two things: A) How much money they have, and B) how much ideological influence they'll have on the rest of the world. That will be a slow process because the financial world is so dominated by the Anglo-American money logic. But even there, some private sector investors are smart enough to know that these orthodox policies do not necessarily produce growth. So they figure out that maybe this BRICS bank is going to do will make the borrowing country grow faster. It's not a completely one-sided game, but the market ideologies are so strong that it's going to be some time before the BRICS bank will have the influence that is commensurate to its financial power.
There has been a lot of news and commentary over Argentina's battle with vulture fund debt holdouts, and a U.S. court's ruling in favor of the vulture funds. Do you think Argentina has pursued a wise course, economically, by first defaulting on its debt, in 2002, and continuing to stand up to the vulture funds?
Well, I really hope it keeps fighting, because it's really making people reassess the nature of the global financial system. The ideal scenario would be that this makes countries agree that we need to introduce some kind of international bankruptcy law. Because in domestic bankruptcy, the exact proportion differs across countries, but typically if 75% to 80% of the creditors agree to have a haircut, then the remaining 20% or 25% have to accept it. They cannot sue the company later. But in the international case, there's no such law, which is what allows these vulture funds to play this game.
So [economist] Mark [Weisbrot] was saying that–because it's not realistic to talk about getting the international bankruptcy law—now countries will want to issue sovereign bonds in countries like Belgium or Britain where that kind of thing wouldn't be allowed. This has become such a scandal that even Martin Wolf of the Financial Times is now on the side of the Argentinians. So yes, it's tough for Argentina, but they have to fight this for the rest of the world.
In your latest book, Economics: The User's Guide, you proclaim that there is no "right" way to do economics, that no school of economic thought has all the answers – and you systematically describe the strong points and shortcomings of each major school of economic thought. Are we witnessing a new era, where this idea is becoming more widely accepted, perhaps as in Latin America? Do you talk to economic policy makers who are deliberately taking ideas that work from different schools and applying them in practice?
Yeah, you know of course some policy makers, especially the ones you found in many Latin American countries in the 1990s were totally driven by ideology – the same story as some of the Korean policy makers in the late '80s and early '90s. On the other hand, many policy makers are very pragmatic. They'll do whatever policy they think is right without really bothering about whether the idea came from Keynes, or Marx, or Adam Smith. So in that sense we actually have more hope with policy makers because professional economists are hide-bound by their ideology. Many of them would not accept some argument not because they didn't agree but because it doesn't fit into their ideology. Policy makers are less like that, so we have more hope with them.
Indeed, Latin America itself has great intellectual traditions in economics – the Latin American structuralist school, dependency theory even. Especially if you go to countries like Brazil you have people doing research with many different theoretical heritages. Brazil has lots of post-Keynesians. At the moment the BNDES is run by these academic industrial economists who are influenced by Schumpeter and the Latin American structuralist school. So they think very differently.
In that sense, the region doesn't lack those alternative traditions to draw from, and this in a way is why I am predicting the region will move further and further away from the Washington Consensus, because they can generate their own, new economic ideas. Of course, subject to the political configurations, subject to external conditions – I'm not saying this is totally predictable. But having seen the development of some of these orthodox policies, they can now kind of push forward much. Of course, it depends on the country. Some countries do not have enough people with this kind of economic background, but one thing that the Latin Americans are good at is continental solidarity. So I'm sure they can help each other, with countries like Brazil and Argentina lending some economists to help Paraguay, or wherever those capacities are lacking.
Dan Beeton is International Communications Director at the Washington-based Center for Economic and Policy Research (CEPR). Ha-Joon Chang is a CEPR Senior Research Associate.
Washington Consensus
Pink Tide


It happened later than I originally intended it to, but the draw for the contest to win a print copy of Fifty-Word Stories: Volume One has taken place!
I guess I should say who won, then.
I put all the names into a hat, and mixed 'em up good.
Congratulations to sketchlayerJosh, who entered via Twitter! I'll be getting in touch so I can send a book winging its way along to you (Canada Post permitting).
Thank you to everyone who got involved on Twitter and Liked my Facebook page to enter.
The 50-word story contest over at www.GeekingOutAbout.com has wrapped up, and the winners have been announced!
Congratulations to Chris Fries and Leslie Hanna, who won an ebook copy of Fifty-Word Stories: Volume One, and to D.C. McMillen, whose story The 8:15 Train (which you can read below) earned the $25 top prize.
We purposely collided on the 8:15. Before reaching this pivotal moment, we'd spent weeks admiring each other from afar. She laughed, taking the blame for our faux collision. Some small talk, then we played hooky at a nearby hotel. Now I take the 7:45.
It was a great contest, and really difficult to judge. Head over to the GeekingOutAbout post to read some more of the really strong entries.
I posted some thoughts about the Valentine's Day contest over at TimSevenhuysen.com, including what I liked about each of the stories that were posted today.
In total, 22 people submitted 29 stories for the contest, and I really enjoyed receiving and reading them all. I hope you'll all submit more stories in the future, both for other contests and for weekly guest stories.
Also, as a special thank you, if you submitted a story for the contest I've emailed you a thank you note which contains a 50% off coupon for the Volume One ebook bundle, so if you use that coupon before February 21, the bundle is under $2!


text:
Hinds gives £5m to 20 'careers hubs' to boost advice and job 'encounters'
11 Jul 2018, 0:01
Schools in 20 areas of England will have access to funding to train staff to give better careers advice and offer pupils more "encounters" with employers.
Damian Hinds, the education secretary, has announced the locations of 20 new "careers hubs", which will share £5 million in government grants and work with schools to improve advice and guidance for pupils.
It follows a successful pilot of the programme in the north east of England.
Overall, 646 secondary schools, 15 sixth forms and 49 further education colleges will be linked to a hub, with each one involving up to 40 different institutions.
Under recent law changes, schools must do more to enlighten pupils about work and study opportunities, especially less traditional routes.
Schools are already required to give training companies access to their pupils, and from September of this year they must publish details of their careers programmes, as well as having a named "careers leader" in place.
By the end of 2020, schools will also be obliged to offer every pupil at least seven "meaningful encounters" with employers over the course of their school career. They must also meet all eight of the government-endorsed "Gatsby benchmarks" of good careers education.
The careers hubs, which will be set up by existing local enterprise partnerships, are intended to help schools meet those requirements, and will offer bursaries to train careers leaders in schools. Of England's 38 local enterprise partnerships, 30 applied to join the initiative and 20 were successful.
Each hub will employ a lead to coordinate activity and build networks and funding for schools to improve employer encounters. There will also be a "central hub fund" equivalent to £1,000 per school or college, although it is not yet clear how that money will be spent.
The North East Local Enterprise Partnership piloted the careers hub model between 2015 and 2017, with the majority of its schools achieving all eight Gatsby benchmarks. The percentage of the region's young people who were not in education, employment or training (NEET) halved during the period, according to the Social Mobility Commission.
Claudia Harris, chief executive of the Careers and Enterprise Company, said the creation of the hubs was the "next step" on a journey of improvement in careers advice.
"Careers education has come a long way over the past few years. The Gatsby benchmarks have shown us what 'excellent' looks like, and innovative work like the North East Hub pilot has set a model for local delivery."
Hinds said the hubs "will support young people with the right advice to help them make decisions about their future by building better links with employers and providing practical guidance and support to improve the provision of careers advice in schools".
The 20 LEPs selected to run hubs
Cornwall & Isles of Scilly
Leeds City Region
York, North Yorkshire and East Riding
Swindon & Wiltshire
Stoke & Staffordshire
Buckinghamshire Thames Valley
The South East (East Sussex)
Leicester & Leicestershire
Heart of South West (Devon, Plymouth, Somerset and Torbay)
New Anglia (Norfolk and Suffolk)
Janet Downs 12 July 2018
It appears this£5m could be one of the 'piecemeal' approaches to school funding which Robert Halfon, Tory chair of the Education Select Committee, complained about in the Commons on 3 July. Scattering crumbs here and there on a few initiatives won't be sufficient to satisfy the need for adequate school funding (in this case, adequate funding to ensure all pupils receive high-quality careers education and guidance). http://www.localschoolsnetwork.org.uk/2018/07/5m-for-careers-hubs-isnt-enough-to-improve-careers-education-and-guidance


Probably THE definitive guide, if there ever was one, for an overall span of everything one needs to know in starting a Food Truck business, David Weber's 'The Food Truck Handbook' seems to launch itself as the first go-to resource for every suggestion on the subject. 'Start, Grow, and Succeed in the Mobile Food Business' is the moniker given, and from what we can tell through reading through that's exactly what it well help you do.
The text is packed with 34 chapters on basically every subject one can think of regarding the startup, maintenance, running, and growth of a food truck; not to mention how to do it WELL and successfully. 261 pages are filled with everything from picking a Concept to picking a Commissary, forming a Business Plan to finalizing/refining Menu, Raising Money to Watching the Numbers, and Building the Infrastructure to Grow the business or simply Exiting properly. The Introduction section even talks about the History of food trucks, only to then be followed by a full chapter detailing what it REALLY takes to operate a food truck and whether or not you're sure you want to continue down that path.
Well, if you do, you can bet this script will assist you on the way. Not only are the summaries of every topic fully informative and in-depth, no quick 2-sentence blurb that barely says anything, but they're rather up-to-date on current concerns. The Raising Money section discusses Crowdsourcing and Microloans, and there's even a whole chapter devoted to Social Media; in fact, it then has an even more in-depth guide to all the main as an appendix in back. Thus it has yet to be too old for proper use.
A total of 14 different interviews with established food truck owners are spread throughout the book at appropriate intervals, with the latter half piled at the end due to their length. Not only do these provide various fun insights to individual food truck cities, but even further tips and important considerations by those who live food trucks day in and day out. These resources are joined by printed templates of various forms that you will definitely need in day-to-day use and startup planning, such as: Daily Checklists, Time Sheets, Starting Expenses Sheet, a 'Location Journal,' and even a 'Countdown List' of what needs to get done before opening week by week. It even has a whole appendix detailing the food truck regulations for different cities!
Basically put, David Weber lists just about anything and everything that you as an owner need to know and how to do it at every stage of this challenging and rewarding business concept.
Though it keeps standard 'equations' for figuring out and understanding WHAT COSG, Revenue, Breakeven point, etc is, the handbook lacks any solid suggestions for what calculations and percentages one should apply to set their menu prices. Secondly, their secondary appendix full of useful templates is lacking anything in the way of 'outlines' for Business Plan development, though the details of each section IS fully covered in the chapter on it. In particular it would have also been nice to have various templates and examples of the different cost and income sheets used in the Financial Planning section, one of the most laborious and mind-numbing tasks to fill out of all. Finally, the appendix of 'Food Truck Regulation' by city only actually lists the MAJOR food truck ones: Austin, Boston, Chicago, Cleveland, Denver, Los Angeles, Miami, Milwaukee, New York, Philadelphia, Portland, San Fran, Seattle, and Washington. In general it really is an amazingly helpful addition, and can at least help one get an idea on what their city's regulations MIGHT be like, but nonetheless it's only really relevant to owners in those specific areas.
And that's about all we can think of really. Perhaps there's a chance it could go even FURTHER in detail with each section, but compared to other food truck startup resources, other books and even online articles, it's already well-and-beyond one of if not THE most comprehensive and useful all-in-one source out there.
Whether you're an experienced restaurant/food industry professional looking to just make a small change or a complete amateur entrepreneur trying to find their goal, or anything in between, The Food Truck Handbook has something for your needs. Supporting knowledge or teaching the necessities, David Weber has certainly put in the effort to provide a full comprehensive text to help just about anyone in their mobile ventures. We would definitely suggest you buy a copy of THIS, maybe one or two other books like we've listed Here to fully flesh out your complete resource needs, before starting your personal journey in this field.