Disproving a weaker form of Hooley's conjecture

2024/07/01 by Hayani, Mounir
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.01045

Abstract

Hooley conjectured that G(x;q) ≪ xlog q, as soon as q→ +∞, where G(x;q) represents the variance of primes p ≤ x in arithmetic progressions modulo q, weighted by log p. In this paper, we study G_η(x;q), a function similar to G(x;q), but including the weighting factor η((p)/(x)), which has a dampening effect on the values of G_η. Our study is motivated by the disproof of Hooley's conjecture by Fiorilli and Martin in the range q \asymp log log x. Even though this weighting factor dampens the values, we still prove that an estimation of the form G_η(x;q) ≪ xlog q is false in the same range.

Related

• A disproof of Hooley's conjecture
• The distribution of the variance of primes in arithmetic progressions
• Sparser variance for primes in arithmetic progression
• Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions
• The variance of the number of prime polynomials in short intervals and in residue classes