Explicit relations between primes in short intervals and exponential sums over primes

TL;DR

This paper establishes explicit quantitative links between error terms in prime exponential sums and primes in short intervals under RH, connecting to Montgomery's pair-correlation conjecture, advancing understanding of prime distribution.

Contribution

It provides the first explicit relations between error terms in exponential sums over primes and primes in short intervals assuming RH, linking these to Montgomery's conjecture.

Findings

Explicit relations under RH between error terms in prime sums and short interval primes

Connections to Montgomery's pair-correlation conjecture

Insights into prime distribution errors

Abstract

Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.