**Number Theory Seminar**

12:00pm via Zoom

**Brad Rodgers**

Queen’s University

**Squarefrees (and B-frees) in short intervals**

In this talk I will discuss joint work with Ofir Gorodetsky and Sacha Mangerel in which we prove a central limit theorem for counts of squarefree integers in random short intervals. In fact closely related ideas allow one to prove a functional limit theorem — we show that a path formed by counting squarefrees in a random short interval tends to a fractional Brownian motion, a concept I’ll explain in the talk. I hope to also discuss the generalization of squarefrees to B-frees which was first introduced by Erdos and analogous limit theorems for B-frees.

1:00pm via Zoom

**Sandro Bettin**

University of Genova

**Continuity and value distribution of quantum modular forms**

Quantum modular forms are functions *f* defined on the rationals whose period functions, such as *psi(x):= f(x) – x^(-k) f(-1/x)* (for level 1), satisfy some continuity properties. In the case of *k=0*, f can be interpreted as a Birkhoff sum associated with the Gauss map. In particular, under mild hypotheses on psi, one can show convergence to a stable law. If *k* is non-zero, the situation is rather different and we can show that mild conditions on *psi* imply that *f* itself has to exhibit some continuity property. Finally, we discuss the convergence in distribution also in this case. This is a joint work with Sary Drappeau.

1:00pm via Zoom

**Anurag Sahay**

University of Rochester

**Moments of the Hurwitz zeta function on the critical line**

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters *0< α ≤ 1*. We consider the integral moments of the Hurwitz zeta function on the critical line **ℛ**(s)=½. We will focus on rational shift parameters. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet *L*-functions, which leads us into investigating moments of products of *L*-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

**Kyle Pratt**

Oxford University

**A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves**

Erdős, Graham, and Selfridge considered, for each positive integer n, the least value of *t _{n}* so that the integers

*n + 1, n + 2, . . . , n + t*contain a subset the product of whose members with

_{n}*n*is a square. An open problem posed by Granville concerns the size of

*t*, under the assumption of the ABC Conjecture. We discuss recent work, joint with Hung Bui and Alexandru Zaharescu, in which we establish some results on the distribution of tn, including an unconditional resolution of Granville’s problem.

_{n}1:00pm, Shoemaker 219.

**Olivia Beckwith**

Tulane University

**Polyharmonic Maass forms and Hecke series for real quadratic fields**

We study polyharmonic Maass forms and show that they are related to ray class extensions of real quadratic fields. In particular, we generalize work of Lagarias and Rhoades to give a basis for the space of polyharmonic Maass forms for Gamma(N). Modifying an argument of Hecke, we show that twisted traces of cycle integrals of certain depth 2 polyharmonic Maass forms are leading coefficients of Hecke L-series of real quadratic fields. This is joint work with Gene Kopp

**Soumendra Ganguly**

Texas A&M University

**Subconvexity for twisted L-functions on GL(3) × GL(2) and GL(3)**

View Abstract

**Alia Hamieh**

University of Northern British Columbia

**Distribution of Values of Logarithmic Derivatives of L-functions**

View Abstract

**Alex Dunn**

California Institute of Technology

**Bias in cubic Gauss sums: Patterson’s conjecture**

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patter-son (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums rst observed by Kummer in 1846.

One important byproduct of our proof is that we show Heath-Brown’s cubic large sieve is sharp under GRH. This disproves the popular belief that the cubic large sieve can be improved.

An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic

main term.

**Alex Rice**

Millsaps College

**Generalized arithmetic progressions and Diophantine approximation by polynomials**

View Abstract

**Chao Liu**

University of Mississippi

**Sums of sets of abelian group elements**

Let G be an additive abelian group and let S be a subset of G. Let Σ(S) denote the set of elements of G which can be expressed as a sum of a nonempty subset of S. We prove that |Σ(S)| ≥ 1/6k^{2} if 0∉ Σ(S) which improved a well-known result by Olson who proved that |Σ(S)| ≥ 1/9k^{2} in 1976.

**Ayla Gafni**

University of Mississippi

**Partitions into powers of primes** (pdf)

**Felipe Goncalves**

Universität Bonn

**Sign Uncertainty** (pdf)

**Steve Lester**

King’s College London

**Quantum variance for dihedral Maass forms** (pdf)

**Thomas Bloom**

University of Oxford

**Arithmetic progressions in dense sets of integers** (pdf)

**Larry Rolen**

Vanderbilt University

**Periodicities for Taylor coefficients of half-integral weight modular forms** (pdf)

**Ayla Gafni**

University of Mississippi

**The History of the Circle Method** (pdf)

**Rizwanur Khan**

University of Mississippi

**The divisor function in arithmetic progressions** (pdf)

**Thái Hoàng Lê**

University of Mississippi

**Subspaces in difference sets and Mobius randomness** (pdf)

**Micah Milinovich**

University of Mississippi

**The distribution of the zeros of the Riemann zeta-function** (pdf)

**Zhenchao Ge**

University of Mississippi

**Essential Components in F_p[t]** (pdf)

**Tsz Ho Chan**

University of Memphis

**On the congruence equation a + b ≡ c (mod p)** (pdf)

**Rizwanur Khan**

University of Mississippi

**Distribution of mass of automorphic forms** (pdf)

**Alex Rice**

Millsaps College

**New Results on Polynomials in Difference Sets** (pdf)

**Ryo Takahashi**

Nagoya University

**Cohomology annihilators and Jacobian ideals** (pdf)

**Andres Chirre Chavez**

IMPA – Instituto Nacional de Matematica Pura e Aplicada (Brazil)

**Bounding S_n(t) on the Riemann hypothesis** (pdf)

**Anh Lê**

Northwestern University

**Nilsequences and multiple correlations along primes with application to Chowla conjecture** (pdf)

**Sean Sather-Wagstaff**

Clemson University

**Semidualizing modules give a defective Gorenstein defect** (pdf)

**Brent Holmes**

University of Kansas

**On the diameter of dual graphs of Stanley-Reisner rings with Serre (S2) property and Hirsch type bounds on abstractions of polytopes** (pdf)

**Pierre-Yves Bienvenu**

University of Bristol

**A survey of the polynomial method in arithmetic combinatorics** (pdf)

**Habiba Kadiri**

University of Lethbridge

**Explicit results in prime number theory** (pdf)

**Colloquium**

**Florian Enescu**

Georgia State University

**Intersection Algebras** (pdf)

**Steve Lester**

Tel Aviv University

**Zeros of modular forms and quantum unique ergodicity** (pdf)

**Emanuel Carneiro**

IMPA, Rio de Janeiro

**Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function**

This talk lies on the interface of analysis and analytic number theory. I will show how to construct a special reproducing kernel Hilbert space related to the Riemann zeta-function and how one can use this space to obtain bounds for the pair correlation of zeros of the zeta-function, extending classical work of P. X. Gallagher (1985). This is a joint work with V. Chandee, M. Milinovich and F. Littmann.

**Tristan Freiberg**

University of Missouri

**Limit points of the sequence of normalized prime gaps**

Let p_n denote the n-th smallest prime number, and let L denote the set of limit points of the sequence of normalized differences between consecutive primes. We show that for k = 50 and for any sequence of k nonnegative real numbers beta_1 < beta_2 < … < beta_k, at least one of the numbers beta_j – beta_i

belongs to L. It follows that more than 2% of all nonnegative real numbers belong to L.

**Ryan Daleida**

Trinity University

**Making imprimitive Dirichlet characters behave primitively**

Given a Dirichlet character \chi mod q, it is traditional to extend \chi to all of Z/qZ by declaring that \chi(n) = 0 when (n,q) \neq 1. When \chi is primitive (i.e. not induced by a Dirichlet character mod d for some proper divisor d of q), this extension endows the associated Gauss sum and L-function with properties that are lost when \chi is imprimitve. In this talk we will introduce a modification to the traditional extension of imprimitive characters which causes them to behave primitively, in the sense that the relevant properties of the Gauss sum and L-function take on the form usually only associated to primitive characters.

**Nathan Jones**

University of Mississippi

**Probability and elliptic curves (I, II and III)**

This expository lecture will be the (first/second/third) of a short series surveying work of Lang and Trotter from the 1970s. For an elliptic curve y^2 = x^3 + ax + b (with a and b integers) and a prime number p, one may consider the elliptic curve modulo p, i.e. one may consider the equation y^2 congruent to x^3 + ax + b modulo p. In particular, it is of wide interest to understand the number N_p of solutions (x,y) modulo p to this congruence, and how this number N_p varies as the prime p varies. In these lectures, we will use probabilistic notions to make very precise conjectures about some aspects of the variation of N_p with p. This talk will be accessible to graduate students.

**Jim Coykendall**

Clemson University

**An Overview of Factorization: Algebraic and Graphical**

Since about 1990, there has been a large amount of effort devoted to the study of factorization in integral domains (as well as in other structures). Much of this study can be interpreted as an attempt to understand how the multiplicative structure of an integral domain “works” when we do not have unique factorization. A classical example is the class group, the size and complexity of which may be interpreted as a measure of “how far” a (Krull) domain is from being a Unique Factorization Domain.

The aim of this talk will be to give an overview of recent factorization theory. We will highlight some basic definitions, examples, and results. We will also highlight some more recent results that lend themselves to visualizations and have interesting connections to graph theory.

**Tsz Ho Chan**

University of Memphis

**A Look at the Modular Hyperbola** (pdf)

**David Farmer**

American Institute of Mathematics

**Finding and calculating L-functions** (pdf)

**Stephan Baier**

University of Bristol

**Subconvexity bounds for L-functions** (pdf)

** Micah Milinovich**

University of Mississippi

**The Riemann Hypothesis…** (pdf)

**Tsz Ho Chan**

University of Memphis

**Sums of two Squares and Almost Squares** (pdf)

**Ryan Daileda**

Trinity University

**Maximal Class Numbers of CM Number Fields** (pdf)

**Andrew Odlyzko**

University of Minnesota

**Zeros of the Riemann zeta function: computations and implications** (pdf)

** Hung Manh Bui**

Oxford University

**Gaps between consecutive zeros of the Riemann zeta-function** (pdf)

**Zhu Cao**

University of Mississippi

**Integer Matrix Exact Covering Systems and Product Identities for Theta Functions** (pdf)

**Neil Epstein**

University of Michigan

**Closure Operations on Ideals in Commutative Rings** (pdf)

**Zhu Cao**

University of Mississippi

**A new proof of Winquist’s Identity** (pdf)

**Zhu Cao**

University of Mississippi

**A Proof of Lagrange’s Four-Square Theorem** (pdf)

**Micah Milinovich**

University of Mississippi

**Distribution of the Prime Numbers** (pdf)