Aled Walker

Kneser's theorem and primes in arithmetic progressions, or 'A little combinatorics goes a long way'

One of the most celebrated results of analytic number theory is Linnik's theorem on primes in arithmetic progressions. This theorem states that, for all natural numbers q and for all natural numbers b that are co-prime to q, the least prime number that is congruent to b modulo q is at most Cq^L, for some absolute constants C and L. From the work of many authors, most recently Xylouris, we now know that L = 5 suffices. But an explicit valid constant C, though effectively computable, has not so far been established. In this talk we will show how one can indeed get a completely explicit result, with respectable constants, provided one replaces 'prime' by 'product of exactly three primes'. The argument eschews some of the deep number-theoretic issues that affect proofs of Linnik's theorem, concerning Siegel zeros, and focuses instead on a classical tool from additive combinatorics, namely Kneser's theorem. This is joint work with Olivier Ramaré.