Joint Distribution of primes in multiple short intervals

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple disjoint short intervals has a multivariate Gaussian logarithmic limiting distribution with weak negative correlation. As a consequence, we derive short-interval counterparts for many important works in the literature of the Shanks–Rényi prime number race, including a sharp phase transition from all races being asymptotically unbiased to the existence of biased races. Our result remains novel, even for primes in a single moving interval, especially under a quantitative formulation of the linear independence conjecture (QLI).