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

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.

References in corpus (5)

• Primes in short intervals
• Applications of the L-functions ratios conjectures
• The variance of the number of prime polynomials in short intervals and in residue classes
• Autocorrelation of ratios of L-functions
• Statistics of the zeros of $L$-functions and arithmetic correlations

Cited by in corpus (5)

• Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in ${\mathbb{F}}_q[t]$
• Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in ${\mathbb{F}}_q(t)$
• A Refined Conjecture for the Variance of Gaussian Primes Across Sectors
• Variance of sums in short intervals and $L$-functions in ${\mathbb{F}}_q[t]$
• Twin prime correlations from the pair correlation of Riemann zeros