Sieve functions in arithmetic bands

TL;DR

This paper investigates the distribution of sieve functions within short arithmetic bands and explores their applications to correlations and weighted Selberg integrals, advancing understanding in analytic number theory.

Contribution

It introduces new results on the distribution of sieve functions in short arithmetic bands and applies these to correlations and weighted Selberg integrals.

Findings

Distribution results for sieve functions in short arithmetic bands

Applications to correlations of arithmetic functions

Analysis of weighted Selberg integrals

Abstract

An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\ast\mu$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the distribution of $f$ over short {\it arithmetic bands} $\cup_{1\le a\le H}\{n\in(N,2N]: n\equiv a\, (\bmod\,q)\}$, with $H=o(N)$, and give applications to both the correlations and to the so-called weighted Selberg integrals of $f$, on which we have concentrated our recent research.