Joint distribution of primes in multiple short intervals

TL;DR

The paper establishes a multivariate central limit theorem for the weighted counts of primes in multiple short intervals under the Riemann hypothesis and linear independence of zeta-zeros, yielding a Gaussian limit with a covariance matrix $\mathcal{C}$ and explicit weak negative correlations driven by a Coulomb-like term $\Delta(|t_j-t_k|)$. The authors develop a LI-based framework that replaces spectral information with random-phase models for zeros, derive the joint distribution, and apply it to short-interval prime-number races, including a sharp phase transition from unbiased to biased outcomes as the number of intervals grows. They further extend Gaussianity to moving intervals under a quantitative LI conjecture (QLI), broadening the regime in which normal limits hold. Taken together, the results provide a rigorous probabilistic picture of primes in many short windows, offering precise predictions for race biases and laying groundwork for further extensions to moving intervals and larger families of races.

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--Rényi 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).

Theorems & Definitions (75)

• Corollary 1.1
• Remark 1.1
• Remark 2.1
• Definition 2.1
• Proposition 2.1
• Proposition 2.2
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 3.1