Number Theory Student Seminar

This is the home of the OSU Number Theory Student Seminar. We hope to cover topics/techniques/papers of interest in both analytic and algebraic number theory, particularly those that may not be covered in the number theory course sequence. We welcome all who would like to attend.

List of scheduled talks

For the Fall 2017 semester, we will meet every other Tuesday at 5:30PM in MW 154, unless otherwise noted. If you are interested in giving a talk, please contact Alex Beckwith at beckwith.67 (at) osu (dot) edu.

• 5 September: Kevin Nowland

Title: The Pre-Trace Formula and Spherical Functions (click for abstract)

The pre-trace formula is the starting point for many results in analytic number theory, including the Selberg trace formula and sup-norm bounds of Maass cusp forms. I will discuss its derivation for $SL(2,\mathbb{R})$ and the spherical functions used in obtaining estimates from it. Nods toward higher rank groups and representation theory will be given.

• 19 September: Alex Beckwith

Title: Limiting Distributions of Additive Functions (click for abstract)

In 1939, Paul Erdős and Mark Kac showed how the prime divisor counting function $\omega(n)$ is normally distributed in a certain sense. This result was one of the first in what is now called probabilistic number theory. I will discuss the history of this result and provide a sketch of its proof using the Selberg-Delange method. I will also discuss some other recent applications of the Selberg-Delange method.

• 3 October: Yongxiao Lin

Abstract: The Petersson formula, formulated by Iwaniec and used as a delta method by Munshi, will be discussed. We will describe, through an example, how Munshi uses the formula as a tool to establish level-aspect subconvexity bounds for certain $L$-functions. The example we shall discuss will be a Burgess-type bound $$L\left(\frac12,g\otimes \chi\right)\ll q^{\frac38+\varepsilon}$$ for the twisted modular $L$-functions, where $\chi$ is a primitive Dirichlet character modulo $q$, and $g$ is a fixed $GL(2)$ Hecke cusp form.

• 31 October: Fernando Venegas-Pérez

Title: On the Zeros of Eisenstein Series (click for abstract)

A big deal of arithmetical information is coded in highly symmetric objects called modular forms. The Eisenstein series are natural examples these objects and play an important role in the theory. With the help of some basic tools in complex analysis and group theory it is possible to determine the location of the zeros of the Eisenstein series. In this talk I will present the methods involved in this search for zeros.

• 14 November: Dhir Patel

Title: Van der Corput's Method and Exponent Pairs (click for abstract)

The problem of bounding the Riemann zeta function, $\zeta(s)$, has stimulated a lot of work on exponential sums of the type $\sum e(f(n))$. Van der Corput's method is one such technique for estimating these sums. This method gives rise to the theory of exponent pairs. An exponent pair is a 2-tuple $(\chi, \lambda)$ which is used to bound the exponential sum. In this talk, we will outline van der Corput's method and construct such exponent pairs.