Correlations of sieve weights and distributions of zeros

2021/01/12 by Aled Walker, Walker, Aled
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.04418

openalex publication_date 2021/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in short intervals, and also on the so-called `form factor' for the pair correlations of the zeros of the Riemann zeta function. Our bounds ultimately rely on the estimates of Bettin--Chandee for trilinear Kloosterman fractions.

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