Jesse Thorner

A new approach to bounding $L$-functions

Abstract:  An $L$-function is a type of generating
function with multiplicative structure which arises from either an
arithmetic-geometric object (like a number field, elliptic curve,
abelian variety) or an automorphic form.  The Riemann zeta function
$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ is the prototypical example of
an L-function.  While $L$-functions might appear to be an esoteric and
special topic in number theory, time and again it has turned out that
the crux of a problem lies in the theory of these functions.  Many
equidistribution problems in number theory rely on one’s ability to
accurately bound the size of $L$-functions; optimal bounds arise from
the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions
to other $L$-functions.  I will discuss some motivating
equidistribution problems along with recent work (joint with K.
Soundararajan) which produces new bounds for $L$-functions by proving
a suitable “statistical approximation” to the (extended) Riemann