text:
\section{Introduction and statement of results}

The $L^q$ norm of a Dirichlet polynomial $F(s)=\sum_{n\leqslant  N}a(n)n^{-s}$ is defined as
\[\|F\|_q^q=\lim_{T\to\infty}\frac{1}{T}\int_0^T |F(it)|^qdt\]
for $0<q<\infty$.
In this note we are interested in the norms of the Dirichlet polynomial
\[f(s)=\sum_{n\leqslant  N}\frac{1}{n^{1/2+s}}.\]
The norms of this function share similarities with moments of the Riemann zeta function on the half-line, being sometimes referred to as pseudomoments of the Riemann zeta function \cite{CG, BBSSZ}. In the paper \cite{CG}, Conrey and Gamburd showed that for $k\in\mathbb{N}$
\begin{equation}\label{2kth norm}
\|f\|_{2k}^{2k}\sim c_k (\log N)^{k^2}
\end{equation}
as $N\to\infty$ for a specific constant $c_k$. This of course bears a strong resemblance to the Keating--Snaith conjecture for the Riemann zeta function \cite{KS} and in fact, the proof of \eqref{2kth norm} employed random matrix theory.

For the continuous case $0<q<\infty$, the norms of $f$ were investigated by Bondarenko, the author, and Seip in \cite{BHS}. It was shown that when $q>1$, we have the same order of magnitude as Conrey and Gamburd's result i.e.\,
\begin{equation}\label{q>1 norms}\|f\|_q\asymp(\log N)^{q/4},\qquad q>1
\end{equation}
and that the lower bound here holds for all $q>0$, that is
\begin{equation*}\|f\|_q\gg(\log N)^{q/4},\qquad q>0.
\end{equation*}

The situation regarding upper bounds for $q\leqslant  1$ is less satisfactory and somewhat more interesting. Here, the results of \cite{BHS} gave
\begin{equation}\label{small norms}
\|f\|_q\ll\begin{cases}(\log N)^{1/4}\log\log N\,\,\,& q=1\\(\log N)^{1/4}\,\,\,& q<1.\end{cases}
\end{equation}
The proof of these upper bounds relied on estimates for the norm of the partial sum operator $S_N$ which is defined by
\[S_N\bigg(\sum_{n=1}^\infty a(n)n^{-s}\bigg)=\sum_{n\leqslant  N}a(n)n^{-s}.\]
On applying a generalisation of M. Riesz' Theorem due to Helson \cite{Helson}, it can be shown that for Dirichlet series $F(s)$ in $\mathscr{H}^q$ we have
\begin{equation}\label{S_N bounds}
\|S_NF\|_q\leqslant \begin{cases}A_q\|F\|_q& q>1,\\B_q\|F\|_1\,\,\,& 0<q<1\end{cases}
\end{equation}
for some absolute constants $A_q$, $B_q$ (see Lemma 3 of \cite{BHS} and also 8.7.2 and 8.7.6 of \cite{R}). Here, the constants $A_q$, $B_q$ blow up like $|q-1|^{-1}$ as $q\to 1$.

We write $f(s)=S_N(P_N(s))$ where $P_N(s)=\prod_{p\leqslant  N}(1-p^{-1/2-s})^{-1}$ so that $\|f\|_q=\|S_NP_N\|_q$. Since $\|P_N\|_q\asymp (\log N)^{q/4}$ for all $q>0$ (see Proposition \ref{easy prop} below), the bound in \eqref{small norms} for $q<1$ follows immediately by \eqref{S_N bounds}. The bound for the 1-norm follows from observing that $\|f\|_1\leqslant  \|f\|_q$ with $q=1+1/\log \log N$ and noting that
\[\|S_NP_N\|_q\leqslant  A_q \|P_N\|_q\ll (q-1)^{-1}(\log N)^{q/4}\ll (\log N)^{1/4} \log\log N.\]

The bounds in \eqref{S_N bounds} apply to general Dirichlet series. However, in the special and most interesting case where all the coefficients of the Dirichlet series are 1, we can do better.

\begin{thm}
\label{1 norm thm}
 There exists a constant $C_q>0$ such that for $0< q\leqslant  1$,
\begin{equation}\|f\|_q\leqslant (1+o(1))C_q(\log N)^{\alpha_q}(\log\log N)^{\frac{1}{2}-\alpha_q}
\end{equation}
where
\[\alpha_q=\frac{1}{4(2-q)}.\]
In particular
\begin{equation}\|f\|_1 \ll (\log N)^{1/4}(\log \log N)^{1/4}.
\end{equation}
\end{thm}

\begin{rem}The constant $C_q$ can in fact be computed explicitly. We provide an example in equation \eqref{1 constant} below.
\end{rem}

The proof of the above theorem uses the fact that we expect $f(s)$ to be well approximated by the Euler product $P_N(s)$ in the mean. Under this assumption, we can then mollify $f(s)$ appropriately. The theorem will follow from an application of H\"older's inequality along with the following two propositions.

\begin{prop}\label{easy prop}Let $\beta \geqslant  0$ and let
\[J(Y,\beta)=\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1/2+it}}\bigg)^{-1}\bigg|^\beta dt.\]
Then \[J(Y,\beta)\sim a_\beta (e^\gamma\log Y)^{\frac{\beta^2}{4}}\]
as $Y\to\infty$ where
\begin{equation}
\label{c_beta}
a_\beta=\prod_{p}\bigg(1-\frac{1}{p}\bigg)^{\beta^2/4}\sum_{m=0}^\infty \frac{d_{\beta/2}(p^m)^2}{p^m}
\end{equation}
\end{prop}

\begin{prop}
\label{not so easy prop}
Let $a_\beta$ be as above,
\begin{equation}
\label{Y bounds}
\exp((\log\log N)^2)\leqslant  Y\leqslant  \exp\bigg(\frac{\log N}{5\log\log N}\bigg),
\end{equation}
and let
\[I(N,Y)=\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\sum_{n\leqslant  N}\frac{1}{n^{1/2+it}}\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1/2+it}}\bigg)^{1/2}\bigg|^2dt.\]
Then
\begin{equation}
\label{I bound}
I(N,Y)=a_1(e^\gamma\log Y)^{1/4}\bigg(\frac{\log N}{e^\gamma\log Y}+\frac{1}{\pi}\log\log Y+O(1)\bigg).
\end{equation}
\end{prop}

On taking $Y=\exp(\log N/5\log\log N)$ and applying H\"older's inequality in the form
\[\|f\|_q^q\leqslant  I(N,Y)^{\frac{q}{2}}J(Y, q/(2-q))^{1-\frac{q}{2}}\]
gives Theorem \ref{1 norm thm}. If we assume that Proposition \ref{not so easy prop} holds in the range
\[Y\leqslant  \exp(\log N/B\log\log N)\] for some general $B>0$ then we find that
\begin{equation}
\label{1 constant}
\|f\|_1\leqslant  (1+o(1))a_1e^{\gamma/4}\Big(\sqrt{B}e^{-\gamma}+\frac{1}{\pi\sqrt{B}}\Big)^{1/2}(\log N)^{1/4}(\log\log N)^{1/4}.
\end{equation}
We note that this is minimised when $B=e^\gamma/\pi$, and the constant becomes $(4/\pi)^{1/4}a_1$. Our restriction of $B\geqslant  5$ could likely be improved.

One may ask what the true order of $\|f\|_q$ is for $0<q\leqslant  1$. It might be reasonable to expect that $\|f\|_q\ll (\log N)^{q/4}$ in the full range of $q$. Certainly, this would be in analogy with the moments of the Riemann zeta function as proved conditionally on the Riemann hypothesis by Harper \cite{H2}, extending the work of Soundarajan \cite{S}. However, in a recent paper of Bondarenko et. \!al \cite{BBSSZ} (Theorem 6.2) it was shown that for $\alpha>1$ and small $q$, the $q$th norm of the Dirichlet polynomial $\sum_{n\leqslant  N} d_\alpha(n)n^{-1/2-s}$ exceeds the upper bound of $(\log N)^{\alpha^2q/4}$ that one might expect. Thus, the analogy with the Riemann zeta function is not as robust as expected, at least for small $q$.

Very recently, Harper \cite{H1} has obtained sharp bounds on the $q$th norm of the Dirichlet polynomial $\sum_{n\leqslant  N}n^{-it}$ when $0\leqslant  q\leqslant  1$. In particular he has shown that
\[\lim_{T\to\infty}\frac{1}{T}\int_0^T\Big|\sum_{n\leqslant  N}n^{-it}\Big|dt\asymp \frac{\sqrt{N}}{(\log\log N)^{1/4}},\] and in doing so has proved a conjecture of Helson.  It would be interesting to see if his methods may be applied to our situation in order to answer the question on the true growth of $\| f\|_q$.

\section{Proof of the Propositions}

Throughout we let $S(Y)$ denote the set of $Y$-- smooth numbers, that is, the set of positive integers all of whose prime divisors are less than or equal to $Y$.
The proof of Proposition \ref{easy prop} is relatively easy so we give that first.

\begin{proof}[Proof of Proposition \ref{easy prop}] For $\alpha\in\mathbb{R}$ let $d_\alpha(n)$ be the coefficients of $\zeta(s)^\alpha$ so that, in particular, $d_\alpha(p)=\alpha$ for all primes $p$. Then we have
\begin{equation*}
\begin{split}
&\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1/2+it}}\bigg)^{-1}\bigg|^\beta dt
=\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\sum_{n\in S(Y)}\frac{d_{\beta/2}(n)}{n^{1/2+it}}\bigg|^2dt.
\end{split}
\end{equation*}
By Carlson's Theorem \cite{SS},
\begin{equation*}
\begin{split}
\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\sum_{n\in S(Y)}\frac{d_{\beta/2}(n)}{n^{1/2+it}}\bigg|^2dt=&\sum_{n\in S(Y)} \frac{d_{\beta/2}(n)^2}{n}.
\end{split}
\end{equation*}
Alternatively, one may argue this by first truncating the sum at $n=\log T$, say, and then applying the Montgomery--Vaughan mean value Theorem  (see Lemma 2 of \cite{GHK} for example).

 Now,
 \[\sum_{n\in S(Y)} \frac{d_{\beta/2}(n)^2}{n}=\prod_{p\leqslant  Y}\sum_{m=0}^\infty\frac{d_{\beta/2}(p^m)^2}{p^m}=C_\beta(Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-\beta^2/4}\]
where
\begin{equation*}
\begin{split}
C_\beta(Y)=&\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{\beta^2/4}\sum_{m=0}^\infty \frac{d_{\beta/2}(p^m)^2}{p^m}\\
=&\prod_{p}\bigg(1-\frac{1}{p}\bigg)^{\beta^2/4}\sum_{m=0}^\infty \frac{d_{\beta/2}(p^m)^2}{p^m}\prod_{p>Y}\Big(1+O(p^{-2})\Big)\\
=&a_\beta\bigg(1+O\Big(\frac{1}{Y\log Y}\Big)\bigg)
\end{split}
\end{equation*}
The result now follows by Mertens' Theorem.
\end{proof}

For the proof of Proposition \ref{not so easy prop} we start by noting that
\begin{equation}\label{I sum}
\begin{split}
I(N,Y)
=&\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\sum_{n\leqslant  N}\frac{1}{n^{1/2+it}}\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1/2+it}}\bigg)^{1/2}\bigg|^2dt\\
=&\lim_{T\to\infty}\frac{1}{T}\int_0^T \bigg|\sum_{n\leqslant  N}\frac{1}{n^{1/2+it}}\sum_{m\in S(Y)}\frac{d_{-1/2}(m)}{n^{1/2+it}}\bigg|^2dt\\
=&\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)\\n_j\leqslant  N}}\frac{d_{-1/2}(m_1)d_{-1/2}(m_2)}{(m_1n_1m_2n_2)^{1/2}}
\end{split}
\end{equation}
where we have again used Carlson's Theorem in the last line.
We wish to apply Perron's formula in the variables $n_1$ and $n_2$ of this last sum. Consequently, we first look at the associated Dirichlet series.

\begin{lem}
\label{F lem}
Let
\[F(s_1,s_2,Y)=\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{d_{-1/2}(m_1)d_{-1/2}(m_2)}{(m_1m_2)^{1/2}n_1^{1/2+s_1}n_2^{1/2+s_2}}.\]
Then
\begin{equation*}
\begin{split}
F(s_1,s_2,Y)=A(\underline{s})\zeta(1+s_1+s_2)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1+s_2}}\bigg)^{1/2}
\end{split}
\end{equation*}
where $A(\underline{s})=A(s_1,s_2,Y)$ is some product that is absolutely convergent (in the limit $Y\to\infty$) in the regions $\Re(s_1),\Re(s_2)>-1/4$.
\end{lem}
\begin{proof}Since the condition $m_1n_1=m_2n_2$ is multiplicative and so are the coefficients of our sum, we may expand it as an Euler product to see that it is
\begin{equation*}
\begin{split}
&\prod_{p\leqslant  Y}\sum_{\substack{e_1+f_1=\\e_2+f_2}}\frac{d_{-1/2}(p^{e_1})d_{-1/2}(p^{e_2})}{p^{\frac{1}{2}e_1+\frac{1}{2}e_2+(\frac{1}{2}+s_1)f_1+(\frac{1}{2}+s_2)f_2}}\prod_{p> Y}\sum_{f_1=f_2}\frac{1}{p^{(\frac{1}{2}+s_1)f_1+(\frac{1}{2}+s_2)f_2}}\\
=&\prod_{p\leqslant  Y}\bigg(1+\frac{1}{4p}-\frac{1}{2p^{1+s_1}}-\frac{1}{2p^{1+s_2}}+\frac{1}{p^{1+s_1+s_2}}+\cdots\bigg)\prod_{p>Y}\bigg(1-\frac{1}{p^{1+s_1+s_2}}\bigg)^{-1}\\
=&A(\underline{s})\zeta(1+s_1+s_2)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1+s_2}}\bigg)^{1/2}.
\end{split}
\end{equation*}
Explicitly, we have
\begin{equation}\label{arithmetic factor}
A(\underline{s})=\prod_{p\leqslant  Y}\frac{(1-p^{-1})^{1/4}(1-p^{-1-s_1-s_2})}{(1-p^{-1-s_1})^{1/2}(1-p^{-1-s_2})^{1/2}}\sum_{\substack{e_1+f_1=\\e_2+f_2}}\frac{d_{-1/2}(p^{e_1})d_{-1/2}(p^{e_2})}{p^{\frac{1}{2}e_1+\frac{1}{2}e_2+(\frac{1}{2}+s_1)f_1+(\frac{1}{2}+s_2)f_2}}
\end{equation}
and, after expanding everything, this is seen to be absolutely convergent (as $Y\to\infty$) in the stated region.
\end{proof}

As usual, there are some extra technicalities in Perron's formula when $N$ is an integer.  To deal with these, note that the sum in \eqref{I sum} is unaffected if we replace $N$ by the nearest half integer, taking $N+1/2$ if $N\in\mathbb{N}$. Since the asymptotic formula in \eqref{I bound} is also unaffected by such a change, we henceforth assume that $N$ is a half integer.

\begin{lem}\label{perron trunc lem}Let $F(s_1,s_2,Y)$ be as in the above lemma.
Then for $Y\geqslant  2$ and large $T_1\leqslant  T_2\leqslant  N$ we have
\begin{multline}\label{I_1 trunc}
I(N,Y)=\frac{1}{(2\pi i)^2}\int_{c_1-iT_1}^{c_1+iT_1}\int_{c_2-iT_2}^{c_2+iT_2}F(s_1,s_2,Y)N^{s_1+s_2}\frac{ds_2}{s_2}\frac{ds_1}{s_1}\\
+O\bigg(\frac{(\log Y)^{2}(\log N)^3}{T_1^{1/2}}\bigg)
\end{multline}
where $c_1,\,c_2\asymp 1/\log N$.
\end{lem}

\begin{proof}

Let $\chi_{[a,b]}(t)$ denote the characteristic function of the interval $[a,b]$. Then for $n\in\mathbb{N}$
\[\chi_{[1,N]}(n)=\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\bigg(\frac{N}{n}\bigg)^s\frac{ds}{s}+O\bigg(\frac{(N/n)^c}{\max(T|\log(N/n)|,1)}\bigg)\]
for any $c>0$. See the first Lemma of chapter 17 of Davenport's book \cite{D} for example.

Applying this to the variables $n_1$ and $ n_2$ in formula \eqref{I sum} with the choice $c_1,\,c_2\asymp 1/\log N$ gives
\begin{multline}
\label{perron trunc 1}
I(N,Y)=\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{d_{-1/2}(m_1)d_{-1/2}(m_2)}{(m_1n_1m_2n_2)^{1/2}}\\
\times\bigg[\frac{1}{2\pi i}\int_{c_1-iT_1}^{c_1+iT_1}\bigg(\frac{N}{n_1}\bigg)^{s_1}\frac{ds_1}{s_1}+O\bigg(\frac{n_1^{-c_1}}{\max(T_1|\log(N/n_1)|,1)}\bigg)\bigg]\\
\times\bigg[\frac{1}{2\pi i}\int_{c_2-iT_2}^{c_2+iT_2}\bigg(\frac{N}{n_2}\bigg)^{s_2}\frac{ds_2}{s_2}+O\bigg(\frac{n_2^{-c_2}}{\max(T_2|\log(N/n_2)|,1)}\bigg)\bigg].
\end{multline}
After swapping the order of summation and integration, the term arising from the product of the two integrals is seen to be the integral in equation \eqref{I_1 trunc}. This swapping is legitimate since by Lemma \ref{F lem} we are in the domain of absolute convergence of $F(s_1,s_2,Y)$. It remains to estimate the error terms.

Let's first consider a term arising from the product of an integral and an error term. Note that the integrals are
\begin{equation*}
\begin{split}
\frac{1}{2\pi i}\int_{c_j-iT_j}^{c_j+iT_j}\bigg(\frac{N}{n_j}\bigg)^{s_j}\frac{ds_j}{s_j}
\ll &n_j^{-c_j}\int_{-T}^T\frac{dt}{|c_j+it|}=n_j^{-c_j}\bigg(\int_{-T}^{-1}+\int_{-1}^1+\int_1^T\bigg)\frac{dt}{|c_j+it|} \\
\ll& n_j^{-c_j}(\log N+\log T)\\
\ll& n_j^{-c_j}\log N.
\end{split}
\end{equation*}
Hence, the product of an integral and an error term is
\[\ll \log N\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{|d_{-1/2}(m_1)d_{-1/2}(m_2)|}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}\max(T_1|\log(N/n_1)|,1)}.\]
We first apply the bound $|d_{-1/2}(m)|\leqslant  1$.

Now, for the terms with $|\log (N/n_1)|\geqslant  T_1^{-1/2}$ we have
\begin{align}\label{perron error 1}
& \log N\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)\\|\log (N/n_1)|\geqslant  T^{-1/2}}}\frac{1}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}\max(T_1|\log(N/n_1)|,1)}\\
&\leqslant  \frac{\log N}{T_1^{1/2}}\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{1}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}}\nonumber\\
&= \frac{\log N}{T_1^{1/2}}\sum_{\substack{m_1n_1=l\\m_1\in S(Y)}}\frac{\sum_{m_2n_2=l,m_2\in S(Y)}n_2^{-c_2}}{m_1^{1/2}n_1^{1/2+c_1}l^{1/2}}\nonumber\\
&\leqslant  \frac{\log N}{T_1^{1/2}}\sum_{\substack{m_1n_1=l\\ m_1\in S(Y)}}\frac{d(l)}{m_1^{1/2}n_1^{1/2+c_1}l^{1/2}}\nonumber\\
&\leqslant  \frac{\log N}{T_1^{1/2}}\sum_{\substack{m_1,n_1\\ m_1\in S(Y)}}\frac{d(m_1)d(n_1)}{m_1n_1^{1+c_1}}\nonumber\\
&\ll \frac{(\log Y)^2(\log N)^3}{T_1^{1/2}}\nonumber
\end{align}
where we have used the inequality $d(mn)\leqslant  d(m)d(n)$.

For the terms with $|\log (N/n_1)|\leqslant  T_1^{-1/2}$ we have $|n_1-N|\leqslant  ANT_1^{-1/2}$ where $A$ is some positive constant. We write $n_1=N+r$ with $|r|\leqslant  ANT_1^{-1/2}$. Then
\begin{equation*}
\begin{split}
& \log N\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)\\|\log (N/n_1)|\leqslant  T^{-1/2}}}\frac{1}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}\max(T_1|\log(N/n_1)|,1)}\\
&\ll \log N \sum_{|r|\leqslant  ANT^{-1/2}}\sum_{\substack{m_1(N+r)=m_2n_2\\m_j\in S(Y)}}\frac{1}{(m_1m_2)^{1/2}(N+r)^{1/2+c_1}n_2^{1/2+c_2}}\\
&=\log N\sum_{|r|\leqslant  ANT^{-1/2}}\sum_{\substack{m_1(N+r)=l\\m_1\in S(Y)}}\frac{\sum_{m_2n_2=l,m_2\in S(Y)}n_2^{-c_2}}{m_1^{1/2}(N+r)^{1/2+c_1}l^{1/2}}\\
&\leqslant  \log N\sum_{|r|\leqslant  ANT^{-1/2}}\sum_{\substack{m_1\in S(Y)}}\frac{d(m_1(N+r))}{m_1(N+r)^{1+c_1}}\\
&\ll (\log N)(\log Y)^2 \sum_{|r|\leqslant  ANT_1^{-1/2}}\frac{d(N+r)}{(N+r)^{1+c_1}}.
\end{split}
\end{equation*}
Now,
\begin{equation*}
\begin{split}
\sum_{|r|\leqslant  ANT_1^{-1/2}}\frac{d(N+r)}{N+r}=&\sum_{N(1-AT_1^{-1/2})\leqslant  n\leqslant  N(1+AT_1^{-1/2})}\frac{d(n)}{n}
\end{split}
\end{equation*}
which is a short logarithmic average, and hence should be small. Indeed, on applying the formula
\[\sum_{n\leqslant  X}\frac{d(n)}{n}=\frac{1}{2}\log^2 X+c_1\log X +c_0+O(X^{-1/2})\]
we see that
\begin{equation*}
\begin{split}
\sum_{N(1-AT_1^{-1/2})\leqslant  n\leqslant  N(1+AT_1^{-1/2})}\frac{d(n)}{n}\ll \frac{\log N}{T_1^{1/2}}.
\end{split}
\end{equation*}
Applying this gives an error term that is $\ll(\log Y)^2(\log N)^2T_1^{-1/2}$.
To sum up: a term arising from the product of an integral and an error term in \eqref{perron trunc 1} is
\[\ll \frac{(\log Y)^2(\log N)^3}{T_1^{1/2}}.\]

The product of the two error terms in \eqref{perron trunc 1} is given by the sum
\begin{equation*}
\begin{split}
&\sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{|d_{-1/2}(m_1)d_{-1/2}(m_2)|}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}\max(T_1|\log(N/n_1)|,1)\max(T_2|\log(N/n_2)|,1)}\\
&\leqslant \sum_{\substack{m_1n_1=m_2n_2\\m_j\in S(Y)}}\frac{|d_{-1/2}(m_1)d_{-1/2}(m_2)|}{(m_1m_2)^{1/2}n_1^{1/2+c_1}n_2^{1/2+c_2}\max(T_1|\log(N/n_1)|,1)}
\end{split}
\end{equation*}
and as we have already seen, this is $\ll T_1^{-1/2}(\log N)^2(\log Y)^2$.
\end{proof}

We take
\[T_1=T=\exp((\log\log N)^2)\]
and $T_2=2T_1$ in Lemma \ref{perron trunc lem} so that the error term in \eqref{I_1 trunc} is $o(1)$. Hence we may concentrate on the integral
\[\frac{1}{(2\pi i)^2}\int_{c_1-iT}^{c_1+iT}\int_{c_2-2iT}^{c_2+2iT}F(s_1,s_2,Y)N^{s_1+s_2}\frac{ds_2}{s_2}\frac{ds_1}{s_1}\]
where we recall that
\begin{equation*}F(s_1,s_2,Y)=\bigg[\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg]A(\underline{s})\zeta(1+s_1+s_2)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1+s_2}}\bigg)^{1/2}.
\end{equation*}

We first shift the contour in $s_2$ to the line with real part $-\kappa_2=-1/\log Y$. We pick up two simple poles; one at $s_2=0$ and the other at $s_2=-s_1$. As usual, the horizontal contours will give a negligible contribution. With our choice of $\kappa_2$ and the upper bound $Y\leqslant  \exp(\log N/(5\log\log N))$ we have $N^{-\kappa_2}\leqslant  (\log N)^{-5}$ and so the vertical contour on the left will also be seen to be negligible. This is the content of the next lemma.

\begin{lem}
\label{s_2 shift}
Let $T=\exp((\log\log N)^2)$ and let $Y$ satisfy the bounds in \eqref{Y bounds}. Then
\begin{multline}\label{I_2 first shift}
I(N,Y)=\bigg[\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg]\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}\bigg(G(s_1,Y)+H(s_1,Y)\bigg)N^{s_1}\frac{ds_1}{s_1}\\
+O\big((\log N)^{-3/4}\big)
\end{multline}
where
\begin{equation}\label{G}
G(s_1,Y)=\bigg[\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{1/2}\bigg]A(s_1,0,Y)\zeta(1+s_1)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}
\end{equation}
and
\begin{equation}\label{H}
H(s_1,Y)=-\frac{N^{-s_1}}{s_1}A(s_1,-s_1,Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1-s_1}}\bigg)^{1/2}
\end{equation}
\end{lem}
\begin{proof}We consider the integral over $s_2$ as part of the rectangular contour with vertices $c_2-2iT,\,c_2+2iT,\,-\kappa_2+2iT,\,-\kappa_2-2iT$ with $\kappa_2=1/\log Y$. This contour encloses $-s_1$ since $\kappa_2\gg c_1$. Hence,
\begin{equation*}
\begin{split}
&\frac{1}{(2\pi i)^2}\int_{c_1-iT}^{c_1+iT}\int_{c_2-2iT}^{c_2+2iT}F(s_1,s_2,Y)N^{s_1+s_2}\frac{ds_2}{s_2}\frac{ds_1}{s_1}\\
=&\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}\bigg[\mathrm{Res}_{s_2=0}\Big(F(s_1,s_2,Y)\frac{N^s_2}{s_2}\Big)+\mathrm{Res}_{s_2=-s_1}\Big(F(s_1,s_2,Y)\frac{N^s_2}{s_2}\Big)\\
&-\frac{1}{2\pi i}\Big(\int_{c_2+2iT}^{-\kappa_2+2iT}+\int_{-\kappa_2+2iT}^{-\kappa_2-2iT}+\int_{-\kappa_2-2iT}^{c_2-2iT}\Big)F(s_1,s_2,Y)N^{s_2}\frac{ds_2}{s_2}\bigg]N^{s_1}\frac{ds_1}{s_1}.
\end{split}
\end{equation*}
The residue terms at $s_2=0$ and $s_2=-s_1$ give rise to the terms $G(s_1,Y)$ and $H(s_1,Y)$  respectively. Thus it remains to estimate the last three integrals.

For $s_2$ on any of these contours we have the bounds
\[A(s_1,s_2,Y)\ll 1\]
and
\begin{equation*}
\begin{split}
\prod_{p\leqslant  Y}&\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg(1-\frac{1}{p^{1+s_1}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1+s_2}}\bigg)^{1/2}\\
\ll&\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg(1+\frac{1}{p^{1+\sigma_1}}\bigg)^{1/2}\bigg(1+\frac{1}{p^{1+\sigma_2}}\bigg)^{1/2}\\
\ll& (\log Y)^{5/4}
\end{split}
\end{equation*}
since $\sigma_1\asymp1/\log N\ll1/\log Y$ and $\sigma_2\ll 1/\log Y$. Therefore, on the regions of integration under consideration we have
\begin{equation}
\label{F bound}
F(s_1,s_2,Y)\ll |\zeta(1+s_1+s_2)|(\log Y)^{5/4}.
\end{equation}

Consider the integral involving the upper horizontal section. Applying the above bound for $F$ we see that this integral is
\begin{equation*}
\begin{split}\ll&(\log Y)^{5/4}\int_{-T}^{T}\int_{c_2}^{-\kappa_2}|\zeta(1+c_1+\sigma_2+it_1+2iT)|N^{c_1+\sigma_2}\frac{dt_1}{|c_1+it_1|}\frac{d\sigma_2}{|\sigma_2+2iT|}\\
\ll&\frac{(\log Y)^{5/4}}{T}(\log N+\log T)\max_{\substack{\sigma_2\in[-\kappa_2,c_2]\\t_1\in[-T,T]}}|\zeta(1+c_1+\sigma_2+it_1+2iT)|
\end{split}
\end{equation*}
with this second factor arising from splitting the $t_1$-integral over the regions $t_1\ll 1$ and $1\ll |t_1|\leqslant  T$.
Since $t_1+2T\leqslant  3T= 3\exp((\log\log N)^2)$ we have
\[1+c_1+\sigma_2\geqs1-\frac{1}{\log Y}\geqslant  1-\frac{1}{(\log \log N)^2}\geqslant  1-\frac{c}{\log(t_1+2T)}\]
and since $t_1+2T\geqslant  T\gg 1$ we may apply the usual bounds (see Theorem 5.17 of Titchmarsh \cite{T} for example)  to give
\begin{equation}\label{zeta bound}
\max_{\substack{\sigma_2\in[-\kappa_2,c_2]\\t_1\in[-T,T]}}|\zeta(1+c_1+\sigma_2+it_1+2iT)|\ll (\log 3T)^5.
\end{equation}
A similar bound for the lower horizontal integral applies.

For the remaining vertical integral we again apply the bound for $F$ given in \eqref{F bound}. We then have something that is
\begin{equation*}
\begin{split}
\ll (\log Y )^{5/4} \int_{-T}^T\int_{-2T}^{2T} |\zeta(1+c_1-\kappa_2+i(t_1+t_2))|N^{c_1-\kappa_2}\frac{dt_1}{|c_1+it_1|}\frac{dt_2}{|-\kappa_2+it_2|}.
\end{split}
\end{equation*}
If $|t_1+t_2|\ll 1$ then we may bound the zeta function by $1/|c_1-\kappa_2|\ll\log Y$ and in the other case we may use the bound \eqref{zeta bound}. Therefore the above integral is
\begin{equation*}
\begin{split}
\ll &(\log Y)^{5/4} \max(\log Y, (\log T)^5)(\log N+\log T)(\log Y+\log T) N^{-1/\log Y}\\
\ll&(\log N)^{17/4}(\log N)^{-5}\\
\leqslant &(\log N)^{- 3/4}
\end{split}
\end{equation*}
\end{proof}

The integral involving $H(s_1,Y)$ is somewhat trickier so we postpone that till later. As for the integral involving $G(s_1,Y)$, we can  compute it directly with residues to give the following.

\begin{lem}\label{G lem}Let $G(s,Y)$ be given by \eqref{G}. Then
\begin{equation}
\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}G(s,Y)N^{s}\frac{ds}{s}=a_1e^{-\gamma}\bigg(\frac{\log N}{\log Y}+\frac{1}{2}\bigg)+O(1/\log Y)
\end{equation}
where $a_1$ is given by \eqref{c_beta}.
\end{lem}

\begin{proof}
Applying the definition of $G$ we have
\begin{multline*}
\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}G(s,Y)N^{s}\frac{ds}{s}\\
=\bigg[\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{1/2}\bigg]\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}A(s,0,Y)\zeta(1+s)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}N^s\frac{ds}{s}.
\end{multline*}
Again, we shift this contour to the line with real part $-\kappa_1=-1/\log Y$ and encounter a double pole at $s=0$. This gives
\begin{equation*}
\begin{split}
&\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}A(s,0,Y)\zeta(1+s)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}N^s\frac{ds}{s}\\
=&\mathrm{Res}_{s=0}\bigg[A(s,0,Y)\zeta(1+s)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}\frac{N^s}{s}\bigg]\\
&-\frac{1}{2\pi i}\bigg(\int_{c_1+iT}^{-\kappa_1+iT}+\int_{-\kappa_1+iT}^{-\kappa_1-iT}+\int_{-\kappa_1-iT}^{c_1-iT}\bigg)A(s,0,Y)\zeta(1+s)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}N^s\frac{ds}{s}.
\end{split}
\end{equation*}
Applying the usual bounds, as in Lemma \ref{s_2 shift}, the horizontal contours are
\[\ll \frac{(\log Y)^{1/2}(\log T)^{5}}{T}\]
and the vertical contour is
\[\ll N^{-\kappa_1}(\log Y)^{5/2}\ll (\log Y)^{5/2}(\log N)^{-5}\ll (\log N)^{-5/2}.\]
The residue term is given by
\[A(0,0,Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{1/2}\bigg[\log N+\frac{1}{2}\sum_{p\leqslant  Y}\frac{\log p}{p-1}+\gamma+\frac{A^\prime(0,0,Y)}{A(0,0,Y)}\bigg].\]
Since
\[\frac{1}{2}\sum_{p\leqslant  Y}\frac{\log p}{p-1}=\frac{1}{2}\log Y+O(1)\]
we have
\begin{multline*}
\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}G(s,Y)N^{s}\frac{ds}{s}=A(0,0,Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)\bigg[\log N+\frac{1}{2}\log Y+O(1)\bigg]\\+O\bigg(\prod_{p\leqslant  Y}\Big(1-\frac{1}{p}\Big)^{1/2}(\log N)^{-5/2}\bigg).
\end{multline*}
The result now follows on applying Mertens' Theorem and the formula
\begin{equation*}
\begin{split}
A(0,0,Y)=&\prod_p\frac{(1-p^{-1})^{1/4}(1-p^{-1})}{(1-p^{-1})^{1/2}(1-p^{-1})^{1/2}}\sum_{\substack{e_1+f_1=\\e_2+f_2}}\frac{d_{-1/2}(p^{e_1})d_{-1/2}(p^{e_2})}{p^{\frac{1}{2}(e_1+e_2+f_1+f_2)}}\\
&\times\prod_{p>Y}\big(1+O(p^{-2})\big)\\
=&a_1\bigg(1+O\Big(\frac{1}{Y\log Y}\Big)\bigg).
\end{split}
\end{equation*}

\end{proof}

By combining Lemmas \ref{s_2 shift} and \ref{G lem} we have the following formula for $I(N,Y)$:
\begin{equation*}
\begin{split}
I(N,Y)=\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p}\bigg)^{-1/4}\bigg[a_1e^{-\gamma}\bigg(\frac{\log N}{\log Y}+\frac{1}{2}\bigg)+I_2(Y)\bigg]+O(1/(\log Y)^{3/4})
\end{split}
\end{equation*}
where
\[I_2(Y)=-\frac{1}{2\pi i}\int_{c_1-iT}^{c_1+iT}A(s,-s,Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1-s}}\bigg)^{1/2}\frac{ds}{s^2}.\]
Thus, in order to prove Proposition \ref{not so easy prop} it suffices to show that $I_2(Y)= a_1\pi^{-1}\log\log Y+O(1)$.

Note that shifting contours to the far left or right in $I_2(Y)$ will not help since the product over primes gets large in both directions. In any case, the residue at $s=0$ is zero since the integrand is an even function and hence has a zero derivative at $s=0$. Indeed, the integrand is small for $t\leqslant  1/\log Y$ and hence the major contribution should occur away from $s=0$. With this in mind, we first shift the contour to the line with $\Re(s)=0$ to exploit some symmetry. Then for $t\geqs1/\log Y$ we may approximate the product over primes by the zeta function, which is allowed since the products are fairly long and we are on the edge of the critical strip. The major contribution is then seen to come from the region $1/\log Y\leqslant  t\leqslant  1$.

\begin{lem}We have
\[I_2(Y)=\frac{1}{\pi}\int_{1/\log Y}^T A(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{t^2}+O(1)\]
\end{lem}

\begin{proof}
We first shift the line of integration to $c=2/\log Y$ and incur a term that is $o(1)$ from the horizontal sections. We then consider $I_2(Y)$ as part of the rectangular contour with vertices $c-iT,\,c+iT,\,iT,\,-iT$ whose left edge has a semicircular indentation centered at $s=0$ of radius $1/\log Y$.

Thus,
\begin{multline*}I_2(Y)=-\frac{1}{2\pi i}\bigg(\int_{-iT}^{-i/\log Y}+\int_{\substack{s=e^{i\theta}/\log Y\\-\frac{\pi}{2}\leqslant  \theta\leqslant  \frac{\pi}{2}}}+\int_{i/\log Y}^{iT}\bigg)A(s,-s,Y)\\
\times\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1-s}}\bigg)^{1/2}\frac{ds}{s^2}+o(1).
\end{multline*}
The integral over the semicircular arc is seen to be $O(1)$ whilst
\begin{equation*}
\begin{split}&-\frac{1}{2\pi i}\int_{-iT}^{-i/\log Y}A(s,-s,Y)\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+s}}\bigg)^{1/2}\bigg(1-\frac{1}{p^{1-s}}\bigg)^{1/2}\frac{ds}{s^2}\\
=&-\frac{1}{2\pi}\int_{-T}^{-1/\log Y}A(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{(it)^2}\\
=&-\frac{1}{2\pi}\int_{1/\log Y}^TA(-it,it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{(it)^2}\\
\end{split}
\end{equation*}
This is equal to the integral in the upper half plane provided that we can show $A(-it,it)=A(it,-it)$. Now,
\[A(s_1,s_2,Y)=\prod_{p\leqslant  Y}\frac{(1-p^{-1})^{1/4}(1-p^{-1-s_1-s_2})}{(1-p^{-1-s_1})^{1/2}(1-p^{-1-s_2})^{1/2}}\sum_{\substack{e_1+f_1=\\e_2+f_2}}\frac{d_{-1/2}(p^{e_1})d_{-1/2}(p^{e_2})}{p^{\frac{1}{2}e_1+\frac{1}{2}e_2+(\frac{1}{2}+s_1)f_1+(\frac{1}{2}+s_2)f_2}}\]
and hence
\[A(-it,it,Y)=\prod_{p\leqslant  Y}\frac{(1-p^{-1})^{1/4}(1-p^{-1})}{|1-p^{-1-it}|}\sum_{\substack{m\geqslant  0}}\Big|\sum_{n_1n_2=p^m}d_{-1/2}(n_1)n_2^{it}\Big|^2{p^{-m}}\]
and the result follows. Incidentally, this last formula shows that $A(it,-it,Y)$ is positive.
\end{proof}

\begin{lem}We have
\[I_2(Y)=\frac{a_1}{\pi} \log\log Y+O(1)\]
where $a_1$ is given by \eqref{c_beta}
\end{lem}

\begin{proof}
We first note that
\begin{equation}\label{I_2}
\begin{split}
I_2(Y)=&\frac{1}{\pi}\int_{1/\log Y}^TA(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{t^2}+O(1)\\
=&\frac{1}{\pi}\int_{1/\log Y}^{\log Y}A(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{t^2}+O(1)
\end{split}
\end{equation}
since $A(it,-it,Y)\ll 1$ and the product over primes is $\ll \log Y$.
Throughout, we consider the variable $s=\sigma+it$ to be in the range
\begin{equation}
\label{range}
\sigma\geqslant  1,\,\,\,\,\,\qquad1/\log Y\leqslant  t \leqslant  \log Y.
\end{equation}
We now approximate the product over primes by the reciprocal of the zeta function. The first few details of this follow Lemma 3.4 of \cite{HT} but we shall present them anyway for clarity.

Let
\[\zeta(s,Y)=\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^s}\bigg)^{-1}.\]
Then,
\[-\frac{\zeta^{\prime}(s,Y)}{\zeta(s,Y)}=\sum_{p\leqslant  Y}\sum_{m\geqslant  1}\frac{\log p}{p^{ms}}=\sum_{n\leq Y}\frac{\Lambda(n)}{n^s}+\sum_{\substack{p\leqslant  Y,m>1\\p^m>Y}}\frac{\log p}{p^{ms}}.\]
On splitting this last sum over $p\leqslant  \sqrt{Y}$ and $\sqrt{Y}<p\leqslant  Y$ we see that
\[-\frac{\zeta^{\prime}(s,Y)}{\zeta(s,Y)}=\sum_{n\leqslant  Y}\frac{\Lambda(n)}{n^s}+O(Y^{1/2-\sigma}).\]
 Applying the usual arguments of contour integration along with Vinogradov's zero free region we find that
\[\sum_{n\leqslant  Y}\frac{\Lambda(n)}{n^s}=-\frac{\zeta^\prime(s)}{\zeta(s)}+\frac{Y^{1-s}}{1-s}+O\big((\log Y)^{-A}\big)\]
for any positive, fixed $A$.

Therefore, upon integrating over the horizontal line from $\sigma=1$ to $\sigma=\log Y$ we find that
\begin{equation}\label{prod asymp}
\begin{split}
\log\zeta(&1+it,Y)\\
=&\log\zeta(1+it)+\int_{1}^{{\log Y}}\frac{Y^{1-\sigma-it}}{1-\sigma-it}d\sigma-\log\frac{\zeta(\log Y+it)}{\zeta(\log Y+it,Y)}+O\big((\log Y)^{-(A-1)}\big)\\
=&\log\zeta(1+it)+\int_{1}^{{\log Y}}\frac{Y^{1-\sigma-it}}{1-\sigma-it}d\sigma+O\big((\log Y)^{-(A-1)}\big)
\end{split}
\end{equation}

Now, for the integral in \eqref{I_2} over the range $t\in [1,\log Y]$ we use the bound
\begin{equation}
\label{exp int bound}
\int_{1}^{{\log Y}}\frac{Y^{1-\sigma-it}}{1-\sigma-it}d\sigma\ll \frac{1}{|t|\log Y}\ll \frac{1}{\log Y}
\end{equation}
so that
\[\log\zeta(1+it,Y)=\log\zeta(1+it)+O(1/\log Y).\]
Hence, upon exponentiating
\[\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\asymp\frac{1}{|\zeta(1+it)|}.\]
Applying this along with the estimate
\[\frac{1}{|\zeta(1+it)|}\ll \log(2+|t|),\,\,\,\,|t|\geqslant  1\]
 gives
\begin{equation*}
\begin{split}
\int_{1}^{\log Y} A(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{t^2} \ll\int_{1}^{\log Y} \log (2+|t|)\frac{dt}{t^2}
\ll1.
\end{split}
\end{equation*}

Over the range $t\in [1/\log Y,1]$ we use the asymptotic formulae
\[A(it,-it,Y)=A(0,0,Y)+O(t)=a_1+O(t),\]
where $a_1$ is given by \eqref{c_beta}, and
\[\exp\bigg(\Re\int_1^{\log Y}\frac{Y^{1-\sigma-it}}{1-\sigma-it}d\sigma\bigg)=1+O\bigg(\int_1^{\log Y}\frac{Y^{1-\sigma}}{|1-\sigma-it|}d\sigma\bigg),\]
which is valid since by \eqref{exp int bound} the integral in the exponential is $\ll 1$ for $t\geqslant  1/\log Y$. We also use the formula
\[\frac{1}{|\zeta(1+it)|}=t+O(t^2).\]
Putting these together via \eqref{prod asymp} gives
\begin{equation*}
\begin{split}
\frac{1}{\pi}&\int_{1/\log Y}^1 A(it,-it,Y)\bigg|\prod_{p\leqslant  Y}\bigg(1-\frac{1}{p^{1+it}}\bigg)\bigg|\frac{dt}{t^2}\\
=&\frac{1}{\pi}\int_{1/\log Y}^1 A(it,-it,Y)\frac{1}{|\zeta(1+it)|}\exp\bigg(\Re\int_1^{\log Y}\frac{Y^{1-\sigma-it}}{1-\sigma-it}d\sigma\bigg)\\
&\qquad\times\Big(1+O((\log Y)^{-A-1})\Big)\frac{dt}{t^2}\\
=&\frac{a_1}{\pi}\int_{1/\log Y}^1\frac{dt}{t}+O\bigg(\int_{1/\log Y}^1\int_1^{\log Y}\frac{Y^{1-\sigma}}{|1-\sigma-it|}d\sigma\frac{dt}{t}\bigg)+O(1)\\
=&\frac{a_1}{\pi}\log\log Y+O\bigg(\log Y\int_1^{\log Y}{Y^{1-\sigma}}d\sigma\bigg)+O(1)\\
=&\frac{a_1}{\pi}\log\log Y+O(1).
\end{split}
\end{equation*}

\end{proof}

\section{Acknowledgments}

The author would like to thank Andrew Granville and Kristian Seip for their useful comments and suggestions on a draft of this paper.  He would also like to thank Oleksiy Klurman for several discussions on this topic.

redpajama_set_name: RedPajamaArXiv


text:
falcon-swagger-ui
=================

|nbsp| |pypi-version| |nbsp|

Simple Falcon application for adding `Swagger UI`_ to your falcon
application.

Included Swagger UI version: v3.38.0

Support Falcon 3.0.0

Installation
------------

``pip install falcon-swagger-ui``

Usage
-----

Example application:

.. code:: python

    import falcon
    from falcon_swagger_ui import register_swaggerui_app

    app = falcon.API()

    SWAGGERUI_URL = '/swagger'  # without trailing slash
    SCHEMA_URL = 'http://petstore.swagger.io/v2/swagger.json'

    ### For developer environment you can expose a static endpoint like:
    # import pathlib
    #
    # SCHEMA_URL = '/static/v1/swagger.json'
    # STATIC_PATH = pathlib.Path(__file__).parent / 'static'
    #
    # @see: http://falcon.readthedocs.io/en/stable/api/api.html#falcon.API.add_static_route
    # app.add_static_route('/static', str(STATIC_PATH))
    #

    page_title = 'Falcon Swagger Doc'
    favicon_url = 'https://falconframework.org/favicon-32x32.png'

    register_swaggerui_app(
        app, SWAGGERUI_URL, SCHEMA_URL,
        page_title=page_title,
        favicon_url=favicon_url,
        config={'supportedSubmitMethods': ['get'], }
    )

Running the example application:

.. code:: bash

    pip install falcon gunicorn jinja2
    gunicorn example:app

Configuration
-------------

The application supports overloading all Swagger UI configuration
options that can be JSON serialized. See
https://github.com/swagger-api/swagger-ui#parameters for options.

In addition, some of the OAuth fields are exposed to special variables
that will be rendered into the relevant function.

Application defaults are listed below (should match SwaggerUI defaults).

.. code:: python

    {
            # OAuth related
            'app_name': 'null',
            'client_realm': 'null',
            'client_id': 'null',
            'client_secret': 'null',

            # SwaggerUI base configuration, see https://github.com/swagger-api/swagger-ui#parameters
            'docExpansion': "none",
            'jsonEditor': False,
            'defaultModelRendering': 'schema',
            'showRequestHeaders': False,
            'supportedSubmitMethods': ['get', 'post', 'put', 'delete', 'patch']
    }

.. _Swagger UI: https://github.com/swagger-api/swagger-ui

.. |bmac-button| image:: https://www.buymeacoffee.com/assets/img/custom_images/yellow_img.png
   :target: https://www.buymeacoffee.com/5xROZDjHE
   :alt: Buy Me A Coffee

.. |pypi-version| image:: https://img.shields.io/pypi/v/falcon-swagger-ui.svg
   :target: https://pypi.org/project/falcon-swagger-ui/
   :alt: PyPI

.. |nbsp| unicode:: 0xA0
   :trim:

redpajama_set_name: RedPajamaGithub


PRESQUE ISLE, Maine - Judy Long, spokesman for Emera Maine, walks us through the steps leading up to, during, and after a power outage. The public is reminded to never touch a downed line or cut tree limbs away from power lines. Make sure generators are properly ventilated as well.


Turkey on Wednesday said neither its airspace nor a strategic air base on its territory were used in the latest US-led strikes against self-proclaimed Islamic State of Iraq and al-Sham (ISIS) insurgents in Syria, a Turkish official said.
"The answer is clear: Neither the (Turkish) airspace nor the Incirlik (air base) were used," the official told AFP in Ankara on condition of anonymity. Incirlik air base in southern Turkey is used by both US and Turkish forces.
Reports based on Syrian activists' accounts had suggested earlier that jets carrying out the latest strikes on ISIS targets inside Syria were seen coming from the Turkish side of the border.
Turkey had so far remained hesitant to play an active role in the US-led coalition to defeat ISIS militants, who were holding dozens of its citizens including diplomats and children.
Their release on Saturday has led to mounting expectations that Turkey will now step up its cooperation with NATO ally the United States.
On Tuesday, President Recep Tayyip Erdoğan welcomed US-led air strikes against the radicals in neighbouring Syria, saying Ankara could provide military or logistical support for the air campaign.
"What the president meant is to be open for all possibilities," the Turkish official told AFP.
"There has been no change in our position," he insisted.


The person who creates the awareness of the problem often gets a lot of attention for creating that awareness. The fact that this person can explain what 's wrong and what needs to be better provides them an audience, especially if they are credible when describing what needs to change. This is an important role, but not the most important role in creating change.
The most important person when it comes to making change happen is the person who is willing to take action to make something different. The person who is willing to own the outcome and lead the change is the critical element in making change. Without someone stepping up and taking responsibility, change doesn't happen. Without the leader of change stepping up and accepting responsibility, all you have is awareness of the problems that exist. This awareness is crucial, but on its own, it doesn't fix anything.
There are a lot of people who can tell you what's wrong. There are many who will be happy to share all of the things that they believe need to change with anyone who will listen. There are far fewer who are willing to step up and lead that change.
The people who make a difference are not the people who talk about change or what might be done to make things better. The people who step up, take action, build consensus, lead the change, and own the outcome are the people who make a difference.
Knowing what needs to change and bringing attention to it is important. So is knowing what has to change for things to be better. But leading change is what makes the difference.


Foodle is a board game about nutrition from South African dietitian and accountant Eddie and Annalien Turner, respectively.
The game contains 300 questions and a board covered with pictures of food. Listen to their promo video if you're a sucker for a South African accent.
The game costs around $46 USD plus shipping, and is available in English and Afrikaans.


Welcome to the Changuinola climate guide, which shows average weather for Changuinola, Panama. Check maximum temperature, the number of days with rainfall, sea temperature and daily hours of sunshine for each month of the year.
The Changuinola climate guide is provided by Weather2Travel.com. For more information about the weather in Changuinola, Panama including monthly rainfall, mininum temperature, UV maximum, heat & humidity and daylight hours visit the Changuinola climate guide page.


Introduction of basic networking principles, protocols and architectures. Course is focused on local area networks (LAN), Ethernet technology, ISO OSI model a general TCP/IP concepts. Because the course is DC oriented, attendees are introduced to various functions of NX-OS operating system for Nexus and MDS platform. Additionally the SAN protocol Fibre Channel and architecture of UCS servers are discussed.
None required; it is an introductory course.
All attendees receive an original of study guide.
Nexus 7000, 5000 and 2000 Series Switches MDS 9000 SAN switches UCS servers with FI 6000 According to starndard two students share one pod.


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Formula 1's 1000th Grand Prix
Posted on April 8, 2019 April 9, 2019 by The Motorsport Critic
This weekend will see Formula 1 contest its one thousandth official Grand Prix. A lot has happened since the very first Grand Prix at Silverstone on 13 May 1950.
This article won't attempt to trace the numerous changes that have taken place since 1950, but will focus on some of the Grand Prix that have taken place, no specific criteria, just some that have popped into my mind.
1950 British Grand Prix
As of writing, with the future of the British Grand Prix uncertain, the very first Formula 1 Grand Prix took place at Silverstone which itself has changed substantially since the inaugural race.
The Grand Prix would be won by Alfa Romeo and the 1950 season would belong to the Italian manufacturer as the then new series is initially dominated by manufacturers, especially from Italy with Alfa Romeo, Ferrari and Maserati all being driven to championship success from 1950 to 1958 with Mercedes also powering Juan Manuel Fangio to the 1954 and 1955 titles.
1960 Indianapolis 500
Perhaps less talked about in Formula 1 is the period when the Indianapolis 500 actively counted towards the championship with it being an official round in Formula 1 from 1950 to 1960.
However due to the different regulations of the two series there was hardly any crossover of teams between the two with only Alberto Ascari and Juan Manuel Fangio making minimal efforts of participating in the Indianapolis 500.
It does however mean that the likes of Bill Vukovich and Jim Rathmann (the winner of the 1960 edition of the 500) aren't just Indianapolis 500 winners but also Formula 1 Grand Prix winners. It means that Bill Vukovich has the same number of Formula 1 Grand Prix victories as Elio de Angelis and more than Jean Alesi and Pastor Maldonado.
1957 German Grand Prix
This marks Juan Manuel Fangio's final Grand Prix victory and also considered one of the greatest victories in the sport after Fangio's disastrous pit stop meant he had to charge through the field, breaking the lap record nine times on his way to victory, around the feared Nordschleife. This is also the race in which Fangio would clinch his fifth world championship.
Fangio would set the standard for anyone wishing to be considered a great within Formula 1 having achieved five world championships, twenty four Grand Prix victories and twenty nine pole positions.
1958 Argentine Grand Prix
This Grand Prix was won by Stirling Moss in a privately entered Cooper. What was significant was that this was the first Grand Prix victory for a rear engined car, a major shift in racing car design, with the series having been contested predominantly with front engined cars up to this Grand Prix.
Cooper themselves would become a force over the next two seasons in the sport, helping Jack Brabham to two titles as smaller teams became more competitive in the sport, challenging established manufacturers for titles.
1966 Belgian Grand Prix
Won by John Surtees, driving for Ferrari, the inclusion of this race is due to Jackie Stewart's accident during the Grand Prix. His ordeal and the subsequent time it took for him to be freed from the car and eventually admitted to hospital would, for better or for worse, make a shift from the danger element in Formula 1 being something that you either lived with or walked away from to a more concentrated effort to ensure that Grand Prix racing became safer.
The focus primarily was on improving the safety of circuits and ensuring that drivers kept themselves safer by wearing fireproof overalls, full face racing helmets and wearing seatbelts. Attempting to slow cars down through technical regulations in order to make racing safer would not really be pursued until the 1990s.
1968 Spanish Grand Prix
Notable for numerous reasons, including being the first Formula 1 race since Jim Clark's death in a Formula 2 race at Hockenheim.
It was also the first time that a major team, Team Lotus, appeared in the colours of their major sponsor, Gold Leaf. Although a privately entered Brabham in the previous round in South Africa had also appeared in sponsors colours. This race and Team Lotus showed a shift from racing in national colours to utilising cars as a way to sell sections of it to prospective sponsors to raise funds.
"Lotus Racing Cars of the 1960s" by David Merrett is licensed under CC BY 2.0
The Gold Leaf Team Lotus livery would kick off introducing some classic sponsor led liveries, including another Team Lotus livery with John Player Special.
The 1968 season saw Formula 1 teams also begin to experiment more with aerodynamics as a way to help improve the performance of the car.
1969 Monaco Grand Prix
Graham Hill wins his fifth Monaco Grand Prix driving for Gold Leaf Team Lotus. Hill's success at Monaco earned him the nickname 'Mr Monaco'.
1976 Japanese Grand Prix
A race that in probability should never have gone ahead but due to the eventful 1976 season, live coverage of the then inaugural Japanese Grand Prix and with a title to be decided as well, it meant that the race went ahead.
This is included partly because of the circumstances in which James Hunt won, having to work his way pass several cars in the closing laps after a late pit stop and of course the retirement of Niki Lauda who, understandably, parked the still healthy Ferrari in the pits due to the atrocious weather conditions.
An enthralling battle near the end of the Monaco Grand Prix between Ayrton Senna and Nigel Mansell eventually sees Senna claim his fifth Monaco Grand Prix victory, equalling the record set by Graham Hill.
Senna's victory broke the run of five consecutive Grand Prix victories that Mansell had taken in the opening five rounds of the 1992 season as Williams would dominate the campaign.
The 1992 Monaco Grand Prix was Senna's first of three victories that the Brazilian would claim during the season.
Senna would go on and claim a sixth Monaco Grand Prix victory in 1993.
1994 San Marino Grand Prix
With Rubens Barrichello's accident on Friday and Roland Ratzenberger's fatal accident on the Saturday, this Grand Prix well and truly reignited the drive to make Formula 1 safer and led to the reforming of the Grand Prix Drivers Association (GPDA).
The Grand Prix and the need to make the sport safer was made relevant further after the fatal accident of Ayrton Senna during the race itself.
The whole weekend would lead to a drive to making changes to both circuits and cars to offer more protection and reduce the risk of serious injury or death in the event of an accident.
2006 Chinese Grand Prix
The last of Michael Schumacher's ninety one Grand Prix victories, he would reset the standards of the levels of dedication required and what could be achieved in Formula 1 after claiming seven world championships and sixty eight pole positions to go with the Grand Prix victories.
His comeback from 2010 to 2012 may not have lived up to many people's expectations, but that has not diminished his impact on the sport and the current generation of drivers within Formula 1 and other forms of motorsport.
2018 Mexican Grand Prix
Lewis Hamilton joins both Juan Manuel Fangio and Michael Schumacher in achieving at least five world championships after finishing fourth in the race.
It also meant that the Mercedes team that Hamilton has driven for since 2013 has remained undefeated in the V6 turbo hybrid era, which started in 2014, emulating what Ferrari achieved with Michael Schumacher in claiming five consecutive drivers and constructors title doubles.
Tags: Ayrton Senna, Formula 1, Indianapolis 500, Juan Manuel Fangio, Lewis Hamilton, Mercedes, Michael Schumacher, Team LotusCategories: Formula 1
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redpajama_set_name: RedPajamaCommonCrawl


Tech Talk: Diff btw 'terminal' and 'console' in Solaris/Linux ?
Diff btw 'terminal' and 'console' in Solaris/Linux ?
A terminal is just a window that people type commands in. A console can be used as aterminal also, but is connected to the system console which is where the kernel andsyslog will output information to.


There's nothing more Australian (or adorable) than a kangaroo and her joey! Hi, I'm kangaroo teddy bear and you get two for the price of one with us as your dynamic duo! I'm extra squashy and am a lifelike brown colour with white paws, inner ears and underbelly. My joey is a tiny mini me who is just as cute (if not cuter)!
Not only would I make a true blue gift for special occasions such as christenings, first birthdays, baptisms and more but being an Aussie icon I'd make an ideal gift to send overseas or as a memorable keepsake from a loved one's trip to Australia. And don't forget the mum-to-be. My little joey makes me just perfect for baby showers – cue the oohs and ahhs when I am unwrapped! Just create your message and it will be embroidered on my tummy for a personalised gift that will last forever.


Click on the button ...This will direct you to my new photography site "NaturalImagesCanada.ca"
This page is showing some of my photos ... Nature always inspires my jewellery designs.
Anna's Hummingbirds .... My year round yard colour!!! Amazing wee creatures!
The Heron got too close to a RW Blackbirds nest and caused quite a commotion!!!


Whilst assessments will be conducted on a case-by-case basis, the proposed focus of assessment for each of the four broad age ranges is outlined in Table 1 below. When conducting assessments, a child and young person's own view of their health and wellbeing status and desired outcomes should be sought. The outcome of each assessment should be shared with the child, inform future assessments, where appropriate, inform placement decisions.


"Jailbreak The Tesla" is a welcome back track by Arizona hip hop trio "Injury Reserve" featuring "Aminé".
The track has been tease since last week When they upload the cover artwork on social media.
Arizona hip hop trio formed in 2013, Injury Reserve fulfill their promise by dropping a track they tease last week.
Though, it was announced that they will drop the track this week and it welcome Aminé.
The group consists of rappers Stepa J. Groggs and Ritchie With a T, and producer Parker Corey.
They joined Ho99o9 and The Underachievers on separate tours in 2017.
In 2018, the group embarked on their first headlining tour and were accompanied by JPEGMAFIA during the second wave of tour dates.
The Arizona trio has been relatively silent for the last year but they're starting off 2019 with a bang.
This could pave way for their new project as it's come alongside the visual.


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Nicolau da Grécia e Dinamarca (22 de janeiro de 1872 - 8 de fevereiro de 1938) foi um membro da Casa de Eslésvico-Holsácia-Sonderburgo-Glucksburgo e o terceiro filho de Jorge I da Grécia, rei dos Helenos, e de Olga Constantinovna da Rússia.

Casamento e descendência
Em 1902, Nicolau casou-se com a grã-duquesa Helena Vladimirovna da Rússia, filha do grão-duque Vladimir Alexandrovich da Rússia e da grã-duquesa Maria Pavlovna da Rússia. Eles tiveram três filhas:

 Olga da Grécia e Dinamarca (11 de junho de 1903 – 16 de outubro de 1997), casada com o príncipe Paulo da Jugoslávia; com descendência.
 Isabel da Grécia e Dinamarca (24 de maio de 1904 – 11 de janeiro de 1955), casada com o conde Carl Theodor de Toerring-Jettenbach; com descendência.
 Marina da Grécia e Dinamarca (13 de dezembro de 1906 – 27 de agosto de 1968), casada com o príncipe Jorge, Duque de Kent; com descendência.

As suas três filhas ficaram famosas por causa da sua beleza.

Vida pública

Juntamente com os seus irmãos Constantino e Jorge, Nicolau ajudou a organizar os Jogos Olímpicos de Verão de 1896 em Atenas, os primeiros a ser realizados desde o ano 393. Nicolau prestou serviço como presidente do sub-comité para o tiro-ao-alvo.

Em 1913, o príncipe assumiu uma posição muito controversa após o assassinato do seu pai em Thessaloniki, declarando que o assassinato do rei tinha sido um instrumento dos serviços secretos alemães, uma teoria que teria ofendido a nova rainha da Grécia, a sua cunhada Sofia da Prússia, que era irmã do kaiser Guilherme II da Alemanha.

O seu pai ofereceu-lhe o Teatro Real da Grécia que Nicolau entregou ao estado grego em 1935. Era amigo de George Simitis e foi padrinho do seu filho, o futuro primeiro-ministro socialista Kostas Simitis.

Morte e enterro
O príncipe Nicolau morreu no Hotel Grande Bretagne em Atenas no dia 8 de fevereiro de 1938 e após uma procissão fúnebre na Catedral Metropoliana de Atenas que contou com a participação de vários membros de sua família, Nicolau foi enterrado no túmulo real no Palácio de Tatoi em 15 de fevereiro de 1938.

Genealogia

Príncipes da Grécia
Príncipes da Dinamarca
Casa de Glücksburg
Gregos do século XIX
Gregos do século XX
Sepultados no Cemitério Real de Tatoi

redpajama_set_name: RedPajamaWikipedia


OSDD approaches drug discovery in a planned yet decentralized manner. The Chief Mentor is Prof. S. K. Brahmachari, Scientist and Founder Director, CSIR- Institute of Genomics & Integrative Biology, Chief Mentor, Open Source Drug Discovery, Academy Professor, Academy of Scientific & Innovative Research, Former Director GeneralCouncil of Scientific and Industrial Research. Prof. M Vijayan, Professor , Indian National Science Academy, Dr. V M Katoch Director General, Indian Council of Medical Research (ICMR), Dr. N.K. Ganguly, Former DG, ICMR, are Mentors. The overall activity is implemented by the OSDD Unit. Each scientific project is executed by a Principal Investigator (PI).


Employees of Telecom Ltd, which was formed after the Government Department of Telecommunication service privatised by the Sri Lankan government recently, are preparing for a general strike, trade union sources said today.
The employees are objecting to a decision made by the Telecom Regulatory Commission which is favourable to other organizations such as Suntel and Lankabel regarding the provision of wireless telephone connections in the commercial areas of Colombo city.
This particular decision has affected the principles of Telecom and the chairman of the company has decided to resign from his post.
Telecom trade union sources said that although this matter was brought to the notice of the Minister of Post and Telecommunications he has failed to find a solution for the problem within the period of time he had mentioned.
"Yesterday was the deadline for the Ministers assurance. But nothing had happened to this minute" a spokesman for Joint Union Front of Telecom said.
He said that their strike would not be a 'traditional' one. "Our action will be secret. The subscribers will feel what we are doing when we go on strike" he added.
Industry sources say the threatened action could disrupt telecommunications across the island.


Angels are private individuals who have access to enough wealth to put into startups and are willing to do so. Tech angels are a particularly rare breed. They may have sold a tech business or may simply be excited by the sector and willing to take a small gamble. However, there are a lot of myths about tech angels and it is important to understand the real situation, or it can be extremely hard to secure funding by this route.
They want to spread their risk – Angel investors are simply people with a lot of money to invest. However, it is important to understand that they will only invest a very small percentage of their wealth into tech startups. It is a risky investment for them and will normally only represent 10%-20% of their overall portfolio. They will spread this portion of their investments over as many as 20 startups over the period in which they invest. Do the sums and be realistic about how much to ask for from each angel.
They respect each other's opinions – Tech angels may want to take the lead or follow an angel that they respect. The lead angel will often be a 'smart' investor who either understands the sector or simply has a track record of investing in tech. This is a useful dynamic to understand. Find a respected lead angel and they will recommend your startup to their friends.
In summary, it is important to understand that founders will not normally find individual angels to fund anything other than very early stage businesses. It's a bit of a numbers game. You may have to find as many as 20 angels to invest in a single funding round. These angels may belong to several networks or may work alone. It is useful to get a commitment from one or two smart investors and get their help in both mentoring and introducing others. Be realistic about the individual investment size and build the total commitment in small chunks. If necessary, take part of the total round and continue fundraising until the total is reached.
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One year after, concessionaires yet to take-over 19 silos
• FG Loses N10.4b Yearly, Delay May Cause Food Crisis
• Hampers Concessionaires' Access To CBN's Credit Facility To Mop Farm Produce
One year after the 19 silo complexes were concessioned by the Federal Government, the concessionaires are yet to take delivery of the silos, The Guardian has learnt.The development might have hampered dreams of ensuring food security in the face of Coronavirus pandemic lockdown, as the current 90,000 metric tonnes (MT) of grains in government's reserve could have been hugely increased.
Apart from this, about N10.4b yearly concession fee and share profit might have been lost by the Federal Government due to the non-release of the silos.From grapevine sources, the current Minister of Agriculture, Sabo Nanono, has threatened to review the contracts, saying he would not hesitate to revoke the licence of any concessionaire that couldn't meet the ministry's terms of agreement.
Checks revealed that the Federal Government could have earned a yearly concession fee of N3.4b and another N6b as profit share of the franchise if the silos had been handed over to the concessionaires early. Unfortunately, the concessionaires still have about N1.3b tied down in the transaction.
But the Federal Government has denied the accusation. The Director, Department of Strategic Grains Reserve, Engr. Sule Haruna, said only three concessionaires, out of the six that won the bids, promptly met the conditions, and they have since been given approval to execute needed repairs.
The concession process, announced on September 19, 2018, by the former Minister of Agric. Audu Ogbeh, after the Federal Executive Council's (FEC) approval, was initiated in 2013 by the then Minister of Agriculture, Akinwumi Adesina, who felt the Federal Government did not have means to manage and maintain the 33 silo complexes.
Adesina, while speaking at the signing of Memorandum of Cooperation (MoC) with the Infrastructure Concession Regulatory Commission (ICRC), said: "Running and managing these huge infrastructure is extremely expensive. At the current guaranteed minimum price rate, about N100b would be required to stock the silos. Government does not have that kind of money to manage them. We, therefore, want the private sector to lease this huge storage infrastructure to improve their management, efficiency and profitability."
Adesina had noted that the MoC was continuation of the process of synergy with the private sector to manage and operate the silos across the country, as they are part of the ministry's storage infrastructure under the Strategic Grains Reserves. He said the network of Strategic Grain Reserve Silo Complexes were established to, among other things, provide immediate food relief in terms of emergencies, provide appropriate mechanism, guarantee minimum price (GMP) scheme to make farmers earn remunerative prices for their produce; provide a mechanism for price stabilisation and storage capacity for excess production and reduce post-harvest losses.
It is, however, unfortunate that seven years after, the ministry is yet to fully put the silos to use. On May 16, 2019, the Federal Government released the silos to six agro allied companies — Agro Universal Consortium, Matriville Nigeria Ltd, Flour Mills Limited, Ebony Agro Industries Ltd, Neon Farms Africa Consortium and Coscharis Farms Ltd, which won the bids, for the benefit of farmers and the nation. Upon conclusion of the transaction process, a ceremonial handover and signing of concession agreement was performed to enable the concessionaires' meet the conditions before physical hand over.
Part of the condition precedents included 25 per cent upfront payment to the Federal Ministry of Agriculture; payment of 75 per cent to the ministry; proof of payment of success fees to the transaction advisers; submission of payment receipt to ICRC; submission of performance bond for review and approval; and submission of insurance cover for review and approval.
Other conditions included proof of representation of warranties; certificate of compliance; proof of no material adverse effect; proof of corporate proceedings, including letters showing board resolutions of the transaction, constituent documents of concessionaires, and letter granting authority to office of the concessionaires executing the concession agreement.
The Guardian checks showed that while five of the concessionaires fulfilled the obligations, only Neon is lagging behind, as it has only made 25 per cent upfront payment to the ministry. Neon Farms got concession for the 25,000mt silo complex capacity in Uyo, Akwa Ibom State.
Findings also revealed that the silo complexes, spread across the geo-political zones, were located at-Ado-Ekiti, Ekiti State (100,000 Metric Tonnes (mt) capacity; Ogoja, Cross River State (25,000mt); Akure, Ondo State (25,00mt); Jos, Plateau State (25,000mt); Sokoto (25,000mt); Gaya, Kano State (25,000mt); Bauchi (25,000mt); Ikenne, Ogun State (25,000mt); Kwali, Federal Capital Territory (100,000mt); and Bulasa, Kebbi State (100,000mt).
Others are Jahun, Jigawa State (25,000mt); Kaduna (25,000mt); Lafiagi, Kwara State (11,000); Makurdi, Benue State (25,000mt); Gombe (25,000mt); Ibadan, Oyo State (25,000mt); Ezillo, Ebonyi State (25,000mt); Uyo, Akwa Ibom State (25,000mt); and Igbariam, Anambra State (25,000mt).
A list indicated that Agro Universal Consortium was awarded eight silos — Ado-Ekiiti, Ogoja, Akure, Jos, Sokoto, Gaya, Bauchi and Ikenne.
Matrixville Consortium was awarded five— Kwali, Bulasa, Jahun, Kaduna and Lafiagi. Flour Mills got three— Markurdi, Gombe and Ibadan. Ebony Agro industries Ltd. was awarded Ezilo, Neon Farms Africa Consortium got Uyo, while Coscharis Farms was awarded the silo complex in Igbariam.
While eight of the silos have been penciled for the next phase of the concession, six would be retained for Federal Government's usage, according to a leaked document from the Ministry of Agriculture.
Though the move was applauded when it was initially announced, considering that the silos had been idle for a long time, after several billions of naira had been expended on their installations, some experts registered their reservations.
Some of them claimed the process was shrouded in secrecy, as farmers were not carried along, insinuating they might have been offered to the then government cronies, who are non-practising farmers or agro processors.
Almost two years after, the concessionaires' investment appears to be going down the drain due to inactivity of the silos.Though in paper, the silos were to be handed-over to the respective concessionaires on May 21, 2019 at the Transcorp Hilton Abuja, according to a letter signed by the Permanent Secretary, FMARD, Dr. Umar Bello Muhammed, dated May 14, 2019 with reference number: FMA/SGR/S/681/IV/24, The Guardian gathered that none of the concessionaires has so far taken possession of the silos despite fulfilling necessary requirements.
"I have the pleasure to invite you to the official handover ceremony of Federal Government of Nigeria Silos to the respective concessionaires taking place at Transcorp Hilton Abuja on the 21st May, 2019.
"The FMARD is implementing a concession process with the objective of partnering with the private sector for the management and operation of silo complexes for the storage of grains in Nigeria and in line with the ERGP, ICRC Act and PPP Policy. The silo concession programme is a win-win situation for the government at all levels, private sectors and the Nigerian farmers," the letter read.
The Director, Department of Strategic Grains Reserve, Haruna, disclosed: "As it is now, Flour Mills, Agro Universal Consortium and Matriville Nigeria Ltd have since completed their repairs and are already test running the plants to enable final handover of the silo complexes under concession to their companies."
Haruna said in contrast, Ebony Agro and Coscharis Farms Ltd only met the condition early February 2020 and subsequently sent request to carry out necessary repairs on March 4, 2020, while Neon Farms is yet to meet the conditions. Some of the concessionaires who narrated their ordeal to The Guardian said they had met all stipulated requirements, including insurance cover for review and approval.
They explained that they were only waiting for the silos to be released to them, but unfortunately, the process was still being held up by the government bureaucracy.
One of them said: "Had the silo complexes been handed over to us in good time, we would have been able to access credit facilities from the Central Bank of Nigeria (CBN) to enable us mop up farm produce from farmers so that we can return to the farm and maintain all-year-round farming cycle. Unfortunately, that has not been done.
"It's a good initiative, but it has to be well implemented. Ogbeh did everything and handed over the certificate to us then. But when the new minister came on board, he wanted to go over all the documentation again. So, the delay is from this new minister."
Private schools preparation for resumption
COVID-19: South Africa hits 500,000 confirmed cases, still not at peak
COVID-19: Lagos Govt extends work-from-home order to Feb 1 January 17, 2021
Italy tightens coronavirus rules, gastronomy largely shut down January 17, 2021
Delayed vaccine delivery to Brazil puts pressure on Bolsonaro January 17, 2021
Any COVID-19 victim that enters Iwo community will be 'automatically' healed ― Oluwo January 16, 2021


text:
package com.linkedin.databus.client.pub;

/**
 * Databus Partition Listener interface. The client application is responsible for implementing a concrete class for this
 * interface and registering in DatabusClient.registerCluster()
 */

public interface DbusPartitionListener
{
	/**
	 *
	 * Listener interface triggered when this client instance starts listening to new partition
	 * Triggered when the registered is created with call-backs, subscriptions and server-side filter set.
	 * Registration will be in INIT state. The client application can use this API to change  checkpoint/regId if needed.
	 *
	 * @param partitionInfo : Databus Partition Info
	 * @param reg : Databus Registration controlling this stream partition.
	 */
	public void onAddPartition(DbusPartitionInfo partitionInfo, DatabusRegistration reg);

	/**
	 * Listener interface triggered when the client instance stops listening to a partition.
	 * Triggered after registration is shutdown but before deregistering.
	 * @param partitionInfo : Databus Partition Info
	 * @param reg : Databus Registration controlling this stream partition.
	 */
	public void onDropPartition(DbusPartitionInfo partitionInfo, DatabusRegistration reg);
}


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text:
Peter Tuchman Net Worth and Income
Peter Tuchman is an American stock exchanger on the New York Stock Exchange who was born in 1958.
Peter Tuchman is also known as the "most photographed trader on Wall Street".
Peter Tuchman estimated net worth as of the year 2022 is $1.5 Billion.
Peter Tuchman is also referred to as the "The Pit Bull of Wall Street" due to his aggressive trading style.
Peter Tuchman is married to Lise Zumwalt Tuchman with whom he had two kids.
Benjamin Tuchman and Lucy Tuchman are the kids of Peter Tuchman.
Shoshana Itzkovich and Marcel Tuchman are the parents of Peter Tuchman.
Peter Tuchman has completed his education in economics and agriculture at the University of Massachusetts.
Peter Tuchman's largest trade till now is the 10 million shares & aims to trade a couple of hundred million dollars of stock.


National Children's Book Week is observed this year from May 7-May 13. You will find everything you need to celebrate books and reading on this site sponsored by the Children's Book Council. Print out the official bookmark, designed by author and illustrator, Lane Smith. See the poster designed by David Wiesner, and perhaps, most important of all, encourage your students to vote for their favorite book. Activities include story starters by many children's authors, and word-search and crossword puzzles. There is a digital toolbox for teachers and librarians, and suggestions on how to make this week a community affair. Celebrate books and the transformative power of reading.


text:
George "Gix" Von Elm (March 20, 1901 – May 1, 1961) was an American professional golfer most noted for his amateur career. He was selected by Golf Digest as Utah's greatest amateur golfer, and in the early 1960s was named Utah Golfer of the Century.

From 1924 to 1931, Von Elm was among the best players in the world. In the 1920s, he worked primarily in the financial and insurance industries, and later designed several golf courses.

Early years
Born in Salt Lake City to Jacob H. and Marie Demmer Von Elm, he began his golf career as a caddie on the old Salt Lake Country Club course, where he was coached by professionals Louis Berrien and Willie Lock. He later refined his game at the Forest Dale Golf Course.

Von Elm attended West High School, where he was an outstanding athlete and played quarterback on the football team. While a 16-year-old high school senior, he won the first of many tournaments, the 1917 Utah Amateur. He won the Utah Amateur again in 1920 and 1921. His golf skills developed quickly, and soon he was seeking competition outside the state. Von Elm attended the University of Utah in Salt Lake City, studying business, and developed his career in insurance and investments.

First significant titles
In 1921, Von Elm won the Pacific Northwest Men's Amateur in a final round that pitted the young Utahn against veteran Chandler Egan in what was described by the Salt Lake Tribune as "one of the most sensational matches in the history of northwest golf." He repeated as winner of that tournament the following year. He captured the 1920 Trans-Mississippi Amateur title, and lost in the finals of that event the next year.

Suspended
Von Elm settled in the Los Angeles area in the early 1920s, but moved back to Utah shortly afterwards, taking a job in a bank. He was suspended by the United States Golf Association from amateur competition for one year, in late 1921, for accepting golf tournament expense money from friends who were Utah golf club members; this was against the rules. Another element of this suspension was his association with the golf manufacturer Spalding, which was against the rules at that time; Von Elm worked as an assistant mining engineer in Colorado for a time.

Returns to form
After serving his suspension, and returning to the Los Angeles area, he joined the Rancho Golf Club, at that time a private facility, and was coached there by pro Arthur Clarkson. He successfully represented Rancho in interclub matches, leading to several team titles in southern California, and won many of the prestigious club invitational titles in the region. Von Elm often played exhibitions and team matches with 1921 British Amateur champion Willie Hunter, also a Rancho member and the club secretary. As his fame grew, he was invited to join such legends as Walter Hagen and Tommy Armour for exhibition and challenge matches, raising significant sums for charity; as an amateur, Von Elm could not accept prize money from these events.

He won the Southern California Amateur three times: 1922, 1925, and 1927. This is a prestigious event which always attracts a strong field. Von Elm won the 1925 Northern California Amateur title, and the 1925 California State Amateur (played at Pebble Beach Golf Links). In 1925 he became the first to win all three major California amateur titles in the same year; this feat has not been repeated since. He finished runner-up in the first Los Angeles Open, held in 1926; this famous tournament is one of the PGA Tour's longest-running events.

Duels with Bobby Jones in U.S. Amateur
In the 1920s amateur golf was in its heyday, with famous players such as Bobby Jones, Chick Evans, and Francis Ouimet. Von Elm's duels with Jones became legendary. In 1923 Von Elm advanced to the quarter-finals of the U.S. Amateur, but lost to former champion Ouimet. In 1924 Von Elm surprised the top golfers by finishing the U.S. Amateur as runner-up to Jones, though Jones won the final match at the Merion Golf Club by a resounding 9 and 8 margin. Von Elm became the first player from west of the Mississippi River to reach the final. The following year Jones eliminated Von Elm in the semifinals, at the Oakmont Country Club. But Von Elm persisted, and in 1926 succeeded in defeating Jones, who had already won the British Open and the U.S. Open that year, to collect the U.S. Amateur title in September at the Baltusrol Golf Club in northeastern New Jersey. A record-breaking crowd of 10,000 erupted with cheers as, in the words of an Associated Press reporter, "The monarch of golf was toppled from his amateur throne...by flaxen-haired George Von Elm...in one of the most stunning upsets of links history." Von Elm, who became the first champion from west of the Mississippi River, was presented with a new car by members of the Rancho Park Golf Course, his home club, upon his return to Los Angeles following the Amateur victory; Jones received a similar gift from Sarasota, Florida golfers that same year, following his wins in the two major Opens.

The win in the national championship capped a superb golf year for Von Elm. Earlier he had played a key role in the American defeat of the British in Walker Cup competition, by halving the final and decisive match over the Old Course at St Andrews, and had tied for third place in the British Open with the famous professional Walter Hagen at Royal Lytham & St Annes Golf Club. Von Elm made two more appearances for his country in the Walker Cup, in 1928 (Chicago Golf Club) and 1930 (Royal St George's Golf Club); he was on the winning U.S. team all three times.

Von Elm lost the longest playoff in the history of the U.S. Amateur when in 1930, he went ten sudden-death, extra holes with Maurice McCarthy Jr. (son of the golf architect and professional of the same name (Metropolitan Amateur champion 1929–30 and 1928 NCAA champion)) at Merion Golf Club, before eventually losing. Von Elm and McCarthy had tied at the end of their regulation 18-hole match.

Golf's longest playoff
Von Elm's most sensational result was as runner-up to pro Billy Burke in the 1931 U.S. Open at the Inverness Club in Toledo, Ohio. After tying Burke with a birdie on the 72nd and final hole of regulation, he competed the next day in a 36-hole playoff, again tying with a birdie on the last hole. This set up a second 36-hole playoff the next day, which Burke won by a stroke. The 72 total holes remains the longest playoff in the history of golf. Burke's victory was the first achieved in a major championship using steel-shafted clubs. Steel shafts had been legalized in the U.S. beginning in 1924. Von Elm had used his normal hickory-shafted clubs in this championship, and complained afterwards that the true essence of the sport of golf was being diminished by the new shaft technology, which took away from traditional shotmaking talent, and made missed shots significantly less penalizing. However, steel shafts were much cheaper and were easier to replace, making the sport much more accessible for a greater range of the population.

Business success
Von Elm ran a successful non-golf business, selling insurance and investments, which generated enough income to fund his amateur golf career during the 1920s. Von Elm stated in 1930, when he turned professional immediately following the U.S. Amateur, that it had cost him an average of $10,000 per year to play his extensive amateur golf schedule; being based in California raised his expenses significantly, since most important American events were held in the East during this era. Von Elm stated that the 1929 stock market crash hurt his business success, and led to his decision to become a businessman golfer the next year, thereby leaving amateur competition, but he retained his business interests.

Enmity with Jones
Although some published sources have indicated a firm friendship between Von Elm and Jones, Von Elm (along with another rival, Chick Evans), had no more than grudging respect for Jones, and in fact resented Jones (for his apparent use of plentiful family money for tournament golf; Jones' father, Robert P. Jones, was a prosperous Atlanta lawyer, while his grandfather, Robert Tyre Jones Sr., was a wealthy Georgia businessman, but it remains unclear how much Jones' relatives assisted with his golf expenses), and questioned his amateur status. Von Elm claimed, without providing proof, that Jones had accepted money for golf films before winning the 1930 U.S. Amateur title in September 1930; Jones did sign a lucrative contract with Warner Brothers to make the films in Hollywood, but this came in November 1930, some two months later, at the same time that he renounced his amateur status. One letter, from after Von Elm's death in 1961, indicated the depth of Von Elm's hatred.

Turns professional
Von Elm turned professional in September 1930, calling himself a 'businessman golfer', and made a career out of his athletic prowess. This was an unusual designation, not followed by any other top amateur players. This declaration made him ineligible for amateur events, including the U.S. Amateur Championship, from that point forward, since accepting money from tournament results ends a player's amateur status.

As a professional, he played out of the Los Angeles area, and won some big-money tournaments, including the 1936 Southern California Open and the 1938 California State Open. The latter event was classified as an official PGA Tour event during the 1920s and 1930s.

Von Elm also remained in the financial services industry, but business became difficult due to the Great Depression.

PGA Tour record
For his record in PGA Tour events, Von Elm is officially credited with five victories, eight second-place finishes, and five third-place finishes, along with 37 finishes in the top ten and 55 finishes in the top 25.

Golf course architect, club professional
Von Elm moved to the Lakeside Golf Club in the late 1930s, after the Internal Revenue Service shut down the original Rancho club. He oversaw the reconstruction of five holes at Lakeside following damage caused by flooding in 1939. In the late 1940s, he consulted to the City of Los Angeles, in the development of an entirely new Rancho club—this time called Rancho Park Golf Course—which became a municipal facility; it was designed and built from scratch by William P. Bell and William Johnson.

From 1950 to 1953, he was the head professional, as well as the head greenskeeper, at the Hacienda CC in California, then moved back to Utah. From 1957 to 1960, he was the professional in eastern Idaho at the golf course in Blackfoot, which he designed and helped to construct during this period. Later he moved to nearby Pocatello, where he directed the design and construction of golf courses in adjacent Alameda and at the Sun Valley resort near Ketchum. According to the important golf course design reference The Golf Course, written by Geoffrey Cornish and Ronald Whitten in 1981, Von Elm is credited with designing the Airport GC in Idaho (1957) and the Mount Ogden municipal course in Utah, the original nine holes at Sun Valley in Idaho, and with remodeling the Hacienda Country Club in California. He is credited with developing the Shadow Mountain club in Palm Springs, California, as well as the Juniper Hills and Highland courses in Idaho.

Family, death, honors, legacy
Von Elm was married three times: Marcella Rodgers (1902–1945), Mary (d.1954), and Billie Dunn (m. 1957). He suffered a severe illness due to an ulcer in 1953, but recovered; he moved back to Utah after this, leaving his job at the Hacienda Club and later lived in eastern Idaho at Blackfoot. After a battle with lung cancer, he died in Pocatello on May 1, 1961.

Von Elm was invited to the Masters Tournament by his great amateur rival Jones in 1951, well past his playing peak, but a significant honor. In 1952, he was named the tenth most important amateur golfer in U.S. history, by a panel of PGA members and sportswriters, marking the half-century list.

Von Elm was inducted in the Charter Class of the Utah Golf Hall of Fame in 1990. He was inducted in 2007 in the inaugural class to the Southern California Golf Hall of Fame.

Two George Von Elm Memorial golf tournaments are played in his honor. From 1962, the Memorial event at the Blackfoot GC in Idaho has been played annually, while the Rancho Park Golf Course, now a municipal facility, in Los Angeles also holds an annual Memorial event named for him.

Amateur wins
this list may be incomplete
1917 Utah Amateur
1920 Trans-Mississippi Amateur, Utah Amateur
1921 Pacific Northwest Amateur,  Utah Amateur
1922 Pacific Northwest Amateur, Southern California Amateur
1925 Southern California Amateur, Northern California Amateur, California State Amateur, Hillcrest C.C. Invitational
1926 U.S. Amateur
1927 Southern California Amateur
1928 Gold Mashie Tournament
1930 French Amateur Championship

Professional wins
this list may be incomplete
1928 Michigan Open (as an amateur)
1936 Southern California Open
1938 California State Open

Major championships

Amateur wins (1)

Results timeline
Amateur

Professional

Note: Von Elm did not play The Open Championship (as a professional) or the PGA Championship.

NYF = Tournament not yet founded
NT = No tournament
CUT = missed the half-way cut
R128, R64, R32, R16, QF, SF = Round in which player lost in match play
"T" indicates a tie for a place

U.S. national team appearances
Amateur
Walker Cup: 1926 (winners), 1928 (winners), 1930 (winners)

General references
Source for U.S. Open & U.S. Amateur:  USGA Championship Database
Source for 1926 British Open:  www.opengolf.com
Source for 1926 British Amateur:  The American Golfer, July, 1926, pg. 58.
Source for 1929 British Open:  www.opengolf.com
Source for 1930 British Open:  www.opengolf.com
Source for 1930 British Amateur:  The Glasgow Herald, May 30, 1930, pg. 13.
Source for 1951 Masters:  www.masters.com

References

External links
Utah Golf Association  - George Von Elm
George Von Elm.com - profile
Utah History to Go  - George Von Elm
GOLF magazine – George Von Elm

American male golfers
PGA Tour golfers
Golf course architects
Golfers from Utah
Golfers from Los Angeles
Golfers from Idaho
University of Utah alumni
Sportspeople from Salt Lake City
Sportspeople from Pocatello, Idaho
1901 births
1961 deaths


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HPU to be developed as 'Model Disabled-Accessible Hill University' in the Country
The Himachal Pradesh University (HPU) is being developed as a "model disabled accessible hill university" in the country, its Vice Chancellor Prof Sikander Kumar said Monday.
The VC was speaking at an awareness workshop on visual impairment and other disabilities on the campus to mark World White Cane Day. World White Cane Day is observed annually on October 15 across the globe to recognise the visually impaired people in the society. The white cane is used by the visually impaired as a tool for safe navigation.
An accessible library has already been made in the university to help achieve this target where visually impaired students read books on computer through a talking software, the VC said at the workshop organised by Ajai Srivastava, member of the HPU Executive Council and Nodal Officer (Disability Affairs). A new section of accessible library with 17 computers will come up this month, whereas the university web portal is already developed as fully accessible to the visually impaired, he added.
Besides, 14 lifts, 10 ramps and accessible toilets on the campus are also on the cards, whereas free education for the differently-abled students, scholarship, 5 per cent in seats and 4 per cent reservation in jobs is already available to them, he added. This year, a record seven disabled HPU students including five visually impaired made the state proud by qualifying the UGC NET, he added.
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He said efforts are on to establish a Centre for Disability and Rehabilitation Studies and one State Disability Resource Centre in the university. HOD Ophthalmology Dr Ram Lal Sharma spoke on the causes of blindness, treatment and cornea transplant.
Several small documentaries on the success of the disabled were also screened and Ajay Kumar, a visually impaired student, made a demo of work on computer through the talking software. Shweta, another visually impaired student, did anchoring and Mohit Kapur read vote of thanks in Braille.


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Jul 29, 2021 @ 4:30pm
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Washington — The House and Senate on Thursday both approved a $2.1 billion supplemental security spending package that directs much-needed funding to the U.S. Capitol Police and National Guard, under strain following the January 6 attack on the U.S. Capitol, and bolsters federal efforts to relocate Afghans who aided U.S. troops during the war in Afghanistan.
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God Only Knows Frequency – There will be hard months, but June 2018 was a fantastic month for the country. Thank you President Obama for lowering profile!
God Only Knows is a song =by the Beach Boys. Sir Paul McCartney has always called this his "favorite song".
didn't Trump outlaw "non-Aryans" in sports?
Oh, right, everything is cool.
There will be tough months – I am a master of the obvious.