On some lower bounds of some symmetry integrals

TL;DR

This paper establishes lower bounds for symmetry and Selberg integrals of arithmetic functions, especially divisor functions, using elementary methods, providing insights into their behavior in short intervals.

Contribution

It introduces new lower bounds for symmetry integrals and derives corresponding bounds for Selberg integrals of divisor functions using elementary techniques.

Findings

Lower bounds for symmetry integrals of arithmetic functions.

Lower bounds for Selberg integrals of divisor functions with k ≥ 3.

Application of elementary methods like Cauchy inequality and Large Sieve bounds.

Abstract

We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some arithmetic functions $f:\N \rightarrow \R$; we obtain from lower bounds of $I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \lq \lq Selberg integral\rq \rq \thinspace of $f$, say $J_f$ (both these integrals give informations about $f$ in almost all the short intervals $[x-h,x+h]$, when $N\le x\le 2N$). In particular, when \thinspace $f=d_k$, the divisor function (having Dirichlet series \thinspace $\zeta^k$, with \thinspace $\zeta$ \thinspace the Riemann zeta function), where $k\ge 3$ is integer, we give lower bounds for the Selberg integrals, say \thinspace $J_k=J_{d_k}$, of the \thinspace $d_k$. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give $I_f$ lower bounds.