James Maynard

Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the
distribution of various arithmetic objects in analytic number theory.
The 'vertical' Sato-Tate law of Katz describes their distribution over a
fixed field F_p, but the equivalent 'horizontal' distribution as the
base field varies over primes remains open. We describe work showing
cancellation in the sum over primes if there are exceptional
Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a
fun blend of ideas from algebraic geometry, the spectral theory of
automorphic forms and sieve theory.