On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals

TL;DR

This paper develops an elementary method to analyze correlations of arithmetic functions, providing bounds for Selberg and symmetry integrals in short intervals beyond classical levels, without using Large Sieve inequality.

Contribution

It introduces a new elementary approach to bound Selberg and symmetry integrals of arithmetic functions in short intervals beyond traditional levels of distribution.

Findings

Non-trivial bounds for Selberg integral in short intervals

Non-trivial bounds for symmetry integral in short intervals

Method avoids Large Sieve inequality, using elementary techniques

Abstract

We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry integral" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the "classical" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary.