Search Results

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 …

The analytic version of Vaughan's identity is $$ \frac{\zeta'}{\zeta} = F+\zeta'G-FG\zeta + \left(\frac{\zeta'}{\zeta}-F\right)(1-\zeta G). $$ Here the last factor to the right is the most complicated …

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by $$ \mathfrak{S}(d_1, \ldots, d_k) (Ck) …

The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …

Whether for a finite set $\mathcal{R}$ of roots the approximation $$ \pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho) $$ is "on average" better …

The maximal domain of meromorphic continuation of a Dirichlet series can be anything. More precisely, for every connected open subset $O$ of $\mathbb{C}$, which contains the half plane $\{\Re s>1\}$, …

Statement a) is true for all $N$. This follows from the fact that the number of even integers which cannot be expressed as the sum of 2 primes is small. The best result in this direction is due to Pin …

In general the answer is no, but if you assume that the $a_n$ are non-negative, then Landau's theorem tells you that $\phi$ has a singularity at $\sigma_{\mathrm{conv}}$, in particular $\sigma_{\mathr …

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 …

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 …

Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that …

The first reason is that you do not gain much by considering major arcs around rational numbers with denominator $\geq 3$. The reason is that the contribution of the major arcs $\{a/q:(a,q)=1\}$ refle …

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) …

No, $\delta_y$ need not tend to 0. Take a rapidly increasing sequence of integers $y_n$. Then define the set $B$ as $\{p-1|\exists n: y_n\leq p\leq 2y_n\}$. Then we have \begin{eqnarray*} \delta_{y_n} …