### 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

Hypothesis.