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.
For the Spring 2018 semester, we will meet every other Tuesday at 5:30PM in MW 154, unless otherwise noted. If you are interested in giving a talk, or would like to be added to the mailing list, please contact Alex Beckwith at beckwith.67 (at) osu (dot) edu.
Upcoming talk

TBD: TBD
Title: TBD (click for abstract)
Abstract: TBD

13 February: Pan Yan
Title: On Unipotent Orbital Integrals for $p$adic Groups (click for abstract)
Abstract: Let $\mathbf{G}$ be a connected reductive linear algebraic group defined over a $p$adic field $F$, and $G=\mathbf{G}(F)$ be its group of $F$rational points. In 1972, R. Ranga Rao showed that the orbital integral $$ \int_{G/{C_G(x)}}f(gxg^{1})dg^* $$ converges for all $x\in G$ and $f\in C_c(G)$. In this talk, I will discuss calculation of unipotent orbital integrals of all spherical functions for $p$adic $SL(2)$ using Rao's formula.

20 February: Yongxiao Lin
Title: Titchmarsh's Argument for Bounding the Epstein Zeta Function (click for abstract)
Abstract: We will describe Titchmarsh's strategy, based on van der Corput's method, for bounding the Epstein zeta function. By using Weyl differencing, a twodimensional second derivative test, Titchmarsh (1934) proved a bound $E\left(\frac{1}{2}+it\right)=O\left(t^{\frac{1}{3}}\log^3 t\right)$, for a particular instance of the Epstein zeta function, $$E(s):=\sum\sum_{(m,n)\in \mathbb{Z}^2\backslash (0,0)} Q(m,n)^{s},$$ where $Q(m,n)=m^2+an^2$ ($a>0$). This is comparable to the classical bound $\zeta\left(\frac{1}{2}+it\right)=O\left(t^{\frac{1}{6}}\log t\right)$ for the Riemann zeta function $\zeta(s)$. If time permits, we will mention briefly recent generalizations of V. Blomer, and of D. Schindler, based on Titchmarsh's argument.

3 April: Rongqing Ye
Title: Gamma factors of cuspidal representations of $GL_n(\mathbb{F}_q)$ (click for abstract)
Abstract: For a pair of cuspidal representations $(\sigma_1, \sigma_2)$ of $GL_n(\mathbb{F}_q)$ and $GL_m(\mathbb{F}_q)$, we define an important invariant $\gamma(\sigma_1 \times \sigma_2)$. We will see how these gamma factors determine a cuspidal representation $\sigma$ via the local converse theorem, and their close connection to those RankinSelberg gamma factors of level zero representations over a local field. At the end of the talk, I will briefly talk about my main thesis topic, exterior square gamma factors.

25 April: Alex Beckwith
Title: Methods for studying the asymptotic behavior of oscillatory integrals (click for abstract)
Abstract: A key step in many typical problems in analytic number theory is to determine the asymptotic behavior of oscillatory integrals of the form $$\int_{\mathbb{R}^d} \omega(x) e^{i \phi(x)} dx,$$ where $\omega$ and $\phi$ are functions smooth functions on $\mathbb{R}^d$ satisfying certain technical conditions. We will discuss some of the various techniques used for studying the asymptotic behavior of such integrals, as well as some general heuristics for determining when one method is advantageous over another. We will also outline the framework developed in a recent preprint of Kiral, Petrow, and Young that greatly simplifies the analysis of certain oscillatory integrals commonly encountered in moment problems in the analytic theory of $L$functions.

5 September: Kevin Nowland
Title: The PreTrace Formula and Spherical Functions (click for abstract)
The pretrace formula is the starting point for many results in analytic number theory, including the Selberg trace formula and supnorm 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 SelbergDelange method. I will also discuss some other recent applications of the SelbergDelange method.

3 October: Yongxiao Lin
Title: PeterssonIwaniecMunshi (click for 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 levelaspect subconvexity bounds for certain $L$functions. The example we shall discuss will be a Burgesstype 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 VenegasPérez
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 2tuple $(\chi, \lambda)$ which is used to bound the exponential sum. In this talk, we will outline van der Corput's method and construct such exponent pairs.