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arXiv · 2401.04000

Joint distribution of primes in multiple short intervals

Sun-Kai Leung
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Abstract

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple short intervals follows a multivariate Gaussian distribution with weak negative correlations. As an application, we obtain short-interval analogues of many results in the literature on the Shanks--R\'enyi prime number race, including a sharp phase transition: biased races between primes in short intervals emerge once the number of intervals exceeds an explicit critical threshold. Our result is new even for a single moving interval, particularly under a quantitative formulation of the linear independence conjecture (QLI).

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Sun-Kai Leung. 2024-01-08. Joint distribution of primes in multiple short intervals. https://arxiv.org/abs/2401.04000

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