On the symmetry of arithmetical functions in almost all short intervals,IV

TL;DR

This paper investigates the symmetry properties of arithmetical functions in short intervals, providing new bounds beyond classical methods by utilizing Weil bounds for Kloosterman sums, applicable in high distribution levels.

Contribution

It introduces a novel approach using Weil bounds to analyze correlations of arithmetical functions in short intervals beyond the classical level.

Findings

Established non-trivial bounds for the Selberg and symmetry integrals

Extended analysis beyond Large Sieve inequality

Achieved results at very high levels of distribution

Abstract

We study the arithmetic (real) function, with f 'essentially bounded'. 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 $[x-h,x+h]$, $N\le x\le 2N$, beyond the 'classical' level, up to a very high level of distribution (for $h$ not too small). This time we go beyond Large Sieve inequality. Precisely, our method applies Weil bound for Kloosterman sums.