Moments of primes in progressions to a large modulus

2024/02/08 by Leung, Sun-Kai
#11N05 #11N13 #60F05 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2402.07941

Abstract

Assuming a uniform q-variant of the prime k-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus q as the residue classes vary. Consequently, depending on the size of φ(q), the prime count follows either a Gaussian or a Poisson distribution. In particular, the least prime in progressions follows an exponential distribution, with some unexpected discrepancies observed for smooth moduli.

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