Event time:

Friday, November 5, 2021 - 4:00pm

Event description:

##### yums seminar series

## Modular Forms, the Sato-Tate Conjecture, and the Riemann Zeta Function

#### Fernando Suarez Trejos

###### Location

TBD - Stay Tuned

###### Abstract

Let p be a prime number. Suppose N(p) is the number of ways to write p as the sum of the squares of 24 integers, i.e., the number of choices {a_1, …, a_{24}}\subset Z for which p = a_1^2+a_2^2+ … +a_{24}^2. It may be shown that N(p) is “approximately” given by N_{approx}(p)=(16/691)(p^{11}+1). How good of an approximation is this, though? As it turns out, this estimate is never “off” by more than (66304/691) p^{11/2}. More interestingly, if you take all the error terms (N(p) -N_{approx}(p)) / p^{11/2} and graph them, it turns out they are distributed exactly in the shape of a semicircle.

The explanation for this phenomenon relates to the concept of modular forms, or more specifically, newforms: complex-analytic functions which encode information relating to very deep problems in number theory. The Sato-Tate conjecture—now a theorem, as of 2011—is essentially the statement that the Fourier coefficients of cuspidal newforms are distributed in the shape of a semicircle. In this seminar, I will discuss modular forms and the history of Sato-Tate, up to and including the recent work of Alexandra Hoey (MIT), Jonas Iskander (Harvard), Steven Jin (UMD), and myself in proving the first unconditional explicit bounds for Sato-Tate. Our discussion will be motivated by the example of the Riemann Zeta function and its applications to the Prime Number Theorem.