On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

TL;DR

This paper links the pair correlation of zeros of L-functions in the Selberg class to the variances of sums of related arithmetic functions over primes in short intervals, extending classical results beyond the Riemann zeta-function.

Contribution

It establishes the equivalence between zero correlation conjectures and variance conjectures for a broad class of L-functions, revealing different variance behaviors based on the degree of the L-functions.

Findings

Variance forms differ for L-functions of degree 2 or higher compared to degree 1.

Two regimes of variance behavior exist for degree ≥ 2, but only one for degree 1.

Results extend classical zeta-function results to a wider class of L-functions.

Abstract

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zeta-function to other $L$-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated $L$-functions is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann zeta-function). Specifically, when the degree is 2 or higher there are two regimes in which the variances take qualitatively different forms, whilst in the degree-1 case there is a single regime.