2020/08/13 by Daniel Fiorilli, Greg Martin, Fiorilli, Daniel +1
Mathematics · #11N13 (11M26) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2008.05837
openalex publication_date 2020/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Define G(x;q) to be the variance of primes p≤ x in the arithmetic progressions modulo q, weighted by log p. Hooley conjectured that as soon as q tends to infinity and x≥ q, we have the upper bound G(x;q) ≪ x log q. In this paper we show that the upper bound does not hold in general, and that G(x;q) can be asymptotically as large as x (log q+logloglog x)2/4.