The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function

TL;DR

This paper explores a generalized Selberg integral and introduces a new pair-correlation function for the zeros of the Riemann zeta-function, linking it to prime number distribution in short intervals.

Contribution

It presents a novel generalized form of the Selberg integral and a new pair-correlation function, enhancing understanding of zeros distribution and prime number patterns.

Findings

Connected the generalized Selberg integral to a new pair-correlation function

Linked the pair-correlation function to prime distribution in short intervals

Provided insights into the zeros of the Riemann zeta-function

Abstract

The present paper is a report on joint work with Alessandro Languasco and Alberto Perelli on our recent investigations on the Selberg integral and its connections to Montgomery's pair-correlation function. We introduce a more general form of the Selberg integral and connect it to a new pair-correlation function, emphasising its relations to the distribution of prime numbers in short intervals.