Table of Contents
Journal of Numbers
Volume 2014, Article ID 140840, 13 pages
http://dx.doi.org/10.1155/2014/140840
Research Article

Generations of Correlation Averages

Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli, Via Cinthia, 80126 Napoli, Italy

Received 5 March 2014; Accepted 28 April 2014; Published 18 June 2014

Academic Editor: Aloys Krieg

Copyright © 2014 Giovanni Coppola and Maurizio Laporta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a general link between weighted Selberg integrals of any arithmetic function and averages of correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions , gauged by the Cesaro weight in the short interval and improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2 th moments of Riemann zeta function on the critical line are derived. We also give general requirements on that allow our treatment for weighted Selberg integrals.