Sun Kai Leung: Joint distribution of primes in multiple short intervals

Abstract:

The study of the distribution of primes in short intervals can be traced back to the time of Gauss. Assuming the prime k-tuple conjecture, Montgomery and Soundararajan showed that primes in a short moving interval follow a normal distribution. The next question we are interested in is: What about primes in multiple short intervals? Are they independent, or, if not, how are they correlated? Assuming the Riemann hypothesis and the linear independence conjecture, we show that the weighted count of primes in multiple short intervals has a multivariate Gaussian logarithmic limiting distribution with weak negative correlation. In this talk, we will also discuss several interesting consequences, which can be regarded as short-interval counterparts of results in the literature of the Shanks—Rényi prime number race.