Lethbridge Number Theory and Combinatorics Seminar: Peng-Jie Wong

Bertrand's postulate states that there is always a prime in the interval [x,2x] for any x≥1. Applying the prime number theorem, one may further show that there is approximately ∫2xxdtlogt primes in [x,2x] for sufficiently large x. There is a more difficult question concerning the distribution of primes p in short intervals when [x,2x] is replaced by [x,x+h] for some h≤x and p is required to be congruent to a modulo q for some (a,q)=1. In this talk, we will discuss how short [x,x+h]
can be. If time allows, we will sketch a proof of the Bombieri-Vinogradov theorem in short intervals, which answers such a question.