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The general approach would be the following: First express $W$ using roots of $\zeta$. Then show that everything converges so well that you can interchange the sum over zeros with the integral over $t …

Explicit estimates for the Moebius function are much harder than for the prime counting function. Ramare (From explicit estimates for the primes to explicit estimates for the Moebius function, Acta Ar …

Under GRH, Titchmarsh showed 1931 that $\sum_{p\leq x}\tau(p+a)\sim C(a)x$, where summation runs over primes only. 1963 Linnik proved the same unconditionally, hence there are many $n$ such that $d(n) …

You cannot do much better then in the construction of Woett. To see this note that the size of $A$ is bounded above by the maximal size of a set $A$, such that $gcd(a, a')<\frac{N}{\log N}$ for all $a …

For an integer $n$ with $1\leq n\leq p-1$, let $n^{-1}$ be the inverse of $n$ modulo $p$. It follows from Weil's bound on Kloosterman sums that for every $\epsilon>0$ the set $\{n: xp\leq n\leq (x+\ep …

As far as I know the Prime Number Theorem in the form $\pi(x+x^\theta)-\pi(x)\sim\frac{x^\theta}{\log x}$ is not proven for any fixed $\theta<\frac{7}{12}$. So I guess the best you could hope for woul …

Define the function $F^*(n)=\underset{p_1\dots p_k\leq n}{\sum_{p_, \ldots, p_k}}\frac{1}{p_1\dots p_k}$. Then we have for each fixed $k$ the asymptotics $F_k^*(n)\sim(\log\log n)^k$. To see this note …

The inequality you state is not a known consequence of GRH, not even in the case $q=1$. In this case von Koch proved 1901 the error term $\mathcal{O}(X^{1/2}\log^2 X)$. Gallagher and Mueller showed th …

It is generally believed that a positive proportion of zeros of $\zeta$ satisfy your condition. In fact, for each fixed $k$, random matrix theory predicts a distribution of the renormalized tuples $(\ …

The best known version of the prime number theorem in short intervals is $\pi(x+y)-\pi(x)\sim\frac{y}{\log x}$, provided that $x^{7/12}<y=o(x)$. So you can cut a connected set $C\subseteq[0,x]^k$ into …

The content of chapter II.9 is contained in many textbooks on analytic number theory. A favourite of mine is Davenport's multiplicative number theory. For binary quadratic forms things are more diffic …

Write $f(x)=\frac{e^{-0.3\sqrt{\log t}}}{\log^2 t}$. On the interval $[2, xf(x)]$ bound the integral trvially by $xf(x)$. On $[xf(x), x]$ the integrand is close to constant. More precisely, we have $$ …

Put $$ H(z) = \sum_{n\geq 1}r(n)z^n = \prod_{n=1}^\infty\left(1-\tau(n)z^n\right)^{-1} = \exp\left(\sum_{n=1}^\infty\sum_{\nu=1}^\infty\frac{\tau(n)z^{n\nu}}{\nu}\right), $$ and $f(t)=H(e^{-t})$. Then …

Studying the distribution of patterns of the Moebius function falls into an easy part, which deals with the distribution of zeroes, and a difficult part, which deals with the distribution of signs. Th …

Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ …