Kneser's theorem and primes in arithmetic progressions, or 'A little combinatorics goes a long way'
20th November 2018, 11:00 am – 12:00 pm
Howard House, 4th Floor Seminar Room
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.