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arXiv · 2402.07941

Moments of primes in progressions to a large modulus

Sun-Kai Leung
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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 $\varphi(q)$, 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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Sun-Kai Leung. 2024-02-08. Moments of primes in progressions to a large modulus. https://arxiv.org/abs/2402.07941

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