Number Theory Seminar

Gene Kopp
Louisiana State University

The Shintani–Faddeev modular cocycle
We ask the question, “how does the infinite q-Pochhammer symbol transform under modular transformations?” and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This “Shintani-Faddeev modular cocycle” is an SL₂(ℤ)-parametrized family of functions generalizing Shintani’s double sine function and Faddeev’s noncompact quantum dilogarithm. We relate real multiplication values of the Shintani–Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.

Jaime Hernandez-Palacios
University Of Mississippi

Gaps between zeros of zeta and L-functions of high degree
There is a great deal of evidence, both theoretical and experimental, that the distribution of zeros of zeta and L-functions can be modeled using statistics of eigenvalues of random matrices from classical compact groups. In particular, we expect that there are arbitrarily large and small normalized gaps between the ordinates of (high) zeros zeta and L-functions. Previous results are known for zeta and L-functions of degrees 1 and 2. We discuss some new results for higher degrees, including Dedekind zeta-functions associated to Galois extensions of $\mathbb{Q}$ and principal automorphic L-functions. This is joint work with Micah Milinovich.

Jesse Thorner
University Of Illinois Urbana-Champaign

A new approach to zero-free regions
I will present a new approach to establishing zero-free regions. The approach is applicable to Rankin-Selberg L-functions for which the method of de la Vallee Poussin is not yet known to be applicable. Applications include the strongest known error term in the prime number theorem for all Rankin-Selberg L-functions. This is joint work with Gergely Harcos.

Antonio Pedro Ramos
SISSA, Italy

The Least Quadratic Nonresidue and the Least Prime in an Arithmetic Progression through Fourier Optimization
In this joint work with E. Carneiro, M. Milinovich and E. Quesada-Herrera, we establish new bounds for the least quadratic nonresidue and the least prime in an arithmetic progression under the Generalized Riemann Hypothesis. We obtain these new bounds by investigating a Fourier extremal problem arising from an application of the Guinand-
Weil explicit formula.

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.

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.

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_n contain a subset the product of whose members with n is a square. An open problem posed by Granville concerns the size of t_n, 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.

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)| ≥ 1/6k² if 0∉ Σ(S) which improved a well-known result by Olson who proved that |Σ(S)| ≥ 1/9k² 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)

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)