Primes in arithmetic progressions on average II

2024/08/31 by Tomos Parry, Parry, Tomos
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.00431

openalex publication_date 2024/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A deep conjecture of Montgomery and Soundararajan on the distribution of prime numbers in short intervals of length h says that the third moment is bounded by ≪ h^\frac 32-c for some c>0. There is in the literature some conditional evidence towards this conjecture whilst in the first article to this series we gave the first instance of unconditional evidence in the form of a bound corresponding to ≪ h^7/5+o(1). In this article we push the exponent down to ≪ h^1+o(1) which more or less is expected to be best possible.

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