Georgia Institute of Technology College of Sciences

I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.

Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod $p$ for various sets of primes $p$; and then sieve methods are the primary tools for doing so. Motivated by the inverse Goldbach problem, Green–Harper, Helfgott–Venkatesh, Shao, and Walsh have explored the inverse sieve problem: if we let $S \subseteq [N]$ be a maximal set of integers in this interval where the residue classes mod $p$ occupied by $S$ have some particular pattern for many primes $p$, what can one say about the  structure of the set $S$ beyond just its size? In this talk, I will give a gentle introduction to inverse sieve problems, and present some progress we made when $S$ mod $p$ has rich additive structure for many primes $p$. In particular, in this setting, we provide several improvements on the larger sieve bound for $|S|$, parallel to the work of Green–Harper and Shao for improvements on the large sieve. Joint work with Ernie Croot and Junzhe Mao.