## Graduate Number Theory Seminar/ Learning Seminar

Co-Organized by Louis Gaudet (louis.gaudet@rutgers.edu) and Brooke Ogrodnik (bl481@math.rutgers.edu, website)

Co-Sponsored by the generosity of the Math Department Academic Initiatives Program (AIP) and the Math Graduate Student Organization (GSO).

If you are interested in giving a talk or would just like to learn please do not hesitate to email the organizer.

The Graduate Number Theory Seminar and Learining Seminar are two seminars that are held periodically throughout the semester.

The Graduate Number Theory Seminar (GNTS) aimed at giving graduate students and postdocs a place to tell others about what they are working on or find interesting, as well as giving them a space and audience to practice giving talks.

The Learning Seminar (LS) currently is a potpourri of topics that interest the speaker. If you have a recommendation for a theme for the Fall semester, please email the organizer. However, overall it is similar to GNTS except faculty are welcome to attend and/or give talks.

#### Next Talk

Date: 4-13-2021 David Herrera The proof of the Prime Number Theorem in Rudin's Functional Analysis. You may have seen any of various proofs of PNT. We will explore the proof of PNT in Walter Rudin's Functional Analysis text. We discuss: The standard transformation of PNT into the asymptotics of Chebyshev's \psi function, Tauberian theorems in general, How PNT follows from Ingham's Tauberian Theorem, Wiener's Theorem and Wiener's Tauberian theorem, and end with a proof of Ingham's Tauberian theorem (and hence PNT). One should obtain the main ideas going into these results and we will discuss as much of the proofs as we can given time constraints.

#### Future Talks

• 4-20 Postdoc Panel

#### Spring 2021- Tuesdays at 4pm ET via Zoom

 April 6th (GNTS): Nathan Mehlhop, "Waring's Problem and the Circle Method" Claire Burrin, "Modular forms of half-integer weight and applications to equidistribution" Max Weinreich, "The pentagram map and arithmetic dynamics" Chris Lutsko, "Pair Correlation of the sequence $\alpha n^\theta \mod 1$" Edna Jones, " An Asymptotic Local-Global Principle for Integral Kleinian Sphere Packings" Forrest Thurman, "The Reproducing Kernel for the Hilbert Space of Modular Forms" Kevin Kwan, "Zeros of Riemann Zeta Function, III" Alex Walker, " Topics in Analytic Continuation" Louis Gaudet, "Critical Zeros of the Riemann Zeta Function" Kim Klinger-Logan, "Pair Correlation for GL(Z) L-Functions"

#### Fall 2020- Wednesdays 1:30pm EDT via Zoom

November 18th (GNTS): Louis Gaudet, "Evaluating a mollified Möbius sum" Brooke Ogronik, "Dirichlet's Class Number Formula" Nick Backes, "On The Theory of p-adic Manifolds" Andre Hernandez-Espiet, "The Rankin-Selberg Method" Kim Klinger-Logan, "Low Energy String Theory and Eisenstein Series" Alex Karlovitz, "Crystallographic Sphere Packings and Local-Global Problems} Edna Jones, "Möbius Transformations and the Bends and Centers of Generalized Circles, Spheres, and Hyperspheres" Louis Gaudet, "Hecke Characters and Applications" George Hauser, "Shimura's "On the Holomorphy of Certain Dirichlet Series" " Alex Walker, "Sets of Large Harmonic Sum Containing no Long Arithmetic Progressions"

#### Summer 2020- Wednesdays 11am via Zoom

August 26th (LS): George Hauser, "Fourier Coefficients of an Eisenstein Distribution on the Universal Cover of SL(2,R)." Edna Jones, "The Local-Global Principle for Integral Soddy Sphere Packings" Brooke Ogrodnik, "On the Local-Global Conjecture for Commutator Traces." Doyon Kim, "Automorphic distributions and the zeros on the critical line pt.2" Alex Karlovitz, "Three Local-Global Problems for Thin Orbits" Surya Gavva, "Prime Number Theorem by Convolution method"

#### Spring 2020- Wednesdays 9:20-10:20 in Hill 423

 May 6th (LS): Doyon Kim, Automorphic distributions and the zeros on the critical line" Max Weinreich, "Automorphism groups of dynamical systems over finite fields" Louis Gaudet, "Vinogradovs theorem on sums of three primes, Part 2," via online seminar Louis Gaudet, "Vinogradovs theorem on sums of three primes, Part 1," VIa Online Seminar Andre Hernandez-Espiet, "Closed Geodesics and Binary Quadratic Forms," Via Online Seminar Alex Karlovitz, "A Theorem on the Base Eigenvalues of Hecke Groups," Via Online Seminar Brooke Ogrodnik, "Local-Global Conjecture for Thin Groups" Claire Burrin, "Elkies-McMullen" Alex Kontorovich, "Part 2 of Ratner's Theorems" Edna Jones, "Local Densities of Diagonal Integral Ternary Quadratic Forms at Odd Primes." Alex Kontorovich, "Ratner's Theorems Part 1" Alex Karlovitz, "Harmonic Analysis on the Hyperbolic Plane" Alex Kontorovich, "A Review on Ratner's Theorems"

#### Fall 2019- Wednesdays 10:30-11:30 in Hill 425

December 11th (GNTS): Forrest Thurman, "Modularity Property of the Theta Functions of Lattices" Alex Karlovitz, "Continuation of Chapter 2 of Ratner's Theorems" George Hauser, "Chapter 2 of Ratner's Theorems" Alex Kontorovich, "Ratner's Theorems" Doyon Kim, "Congruent numbers and elliptic curves" Alex Walker, "Arithmetic Applications of Ratner's Theorems" Edna Jones, "Siegel's Formula" Louis Gaudet, "A zero-density theorem for the Riemann zeta function" Alex Karlovitz, "An Introduction to Maass Forms and their Fourier Expansions" George Hauser, "The Dedekind Eta Function and Its Multiplier System" Matthew Welsh, "The Oppenhiem conjecture and Ratner's theorems" Alex Kontorovich, "Introduction to Ratner's theorems" Brooke Ogrodnik, "Reciprocal Geodesics"