Gaps between Primes in an Arithmetic Progression

Abstract: 

Let tt be a natural number.  We show that there are infinitely many tt-tuples of primes p1<⋯<ptp1<⋯<pt, all congruent to aa modulo qq (with gcd(a,q)=1gcd(a,q)=1 and qq satisfying certain conditions), such that

pt–p1≪qexp(Bt).pt–p1≪qexp⁡(Bt).

Here the value of BB depends on qq.  The proof uses the methods in the recent breakthrough of J. Maynard on the small gaps between primes as well as other inputs from the study of the (possible) Siegel zeros of Dirichlet LL-functions.  This is joint work with R. C. Baker.  Technical details of the work will be kept at a minimum during the talk.