On some lower bounds of some symmetry integrals

Abstract

We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some arithmetic functions $f:\mathbb{N}\to \mathbb{R}$; 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.

arXiv commentPlainTeX(10 p.).Improved results,corrected previous(v1)Lemma (on discrete mean-square,not integral!)

References in corpus (1)

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