Number Theory Seminar

Many important questions ask for showing the existence of primes in `thin' sets  - those with $O(x^{1-\epsilon})$ elements less than $x$. The thinness of such sets presents several analytic difficulties.

The set of numbers with the digit 7 not appearing anywhere in their decimal expansion is a thin set, but has some unusually nice properties which circumvents some (but not all) of these analytic difficulties, and so is a nice test case. We show that there are infinitely many primes in this set. Our proof uses a mixture of bilinear sums, Fourier analysis, geometry of numbers and moment estimates related to a Markov process.