Illinois Number Theory Seminar : Fall 2017

Dong Dong (Illinois)

Polynomial Roth type theorems in Finite Fields

Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.

Sieg Baluyot (Illinois)

On the zeros of Riemann's zeta-function

In the first part of this talk, we present a new proof that a positive proportion of the zeros of the Riemann zeta-function lie on the critical line. The proof is an enhancement of a zero-detection method of Atkinson from the 1940s, and uses the recent estimate of Hughes and Young for the twisted fourth moment of zeta.
In the second part, we consider the number of zeros of zeta inside the region with real part larger than and imaginary part between 0 and T. A bound for this number is called a “zero-density estimate.” We present an improved zero-density estimate for the case when is larger than 1/2 but close to 1/2. The main theorem confirms an unproved result of Conrey from the 1980s using his technique of applying Kloosterman sum estimates.
Finally, in the third part, we look at hypothetical statements for the vertical distribution of zeros along the critical line and deduce their consequences for the prime numbers and other properties of zeta. The main theorem generalizes results of Goldston, Gonek, and Montgomery that give consequences of the pair correlation conjecture. We apply the theorem to examine implications of the well-known “alternative hypothesis,” which is related to Landau-Siegel zeros.

Alexander Dunn (Illinois)

Partition asymptotics and the polylogarithm

In 2015 Vaughn obtained asymptotic formulas for the number of partitions of an integer into squares. Gafni extended this to kth powers. Here we obtain such formulas for the number of partitions into values of an arbitrary integer polynomial $f$ subject to some mild hypotheses. Our methods use an interplay of the circle method, the polylogarithm, and the Matsumoto-Weng zeta function. This is joint work with Nicolas Robles.

George Andrews (Penn State)

4-Shadows in q-Series: Gupta, Kimberling, the Garden of Eden and the OEIS.

This talk is devoted to discussing the implications of a very elementary technique for proving mod 4 congruences in the theory of partitions.  It starts with a tribute to the late Hans Raj Gupta and leads in unexpected ways to partitions investigated by Clark Kimberling, to Bulgarian Solitaire, and to Garden of Eden partitions.  Each surprise busts forth from the OEIS.

Yifan Yang (National Chiao Tung University)

Kyle Pratt (Illinois)

George Shakan (Illinois)

Constructing sets using de Bruijn sequences

Junxian Li and I showed that a set with distinct consecutive r-differences has large sumset. During this talk I will explain how we used de Bruijn sequences to demonstrate that the bound we obtained is tight. If time permits, I’ll talk about a related generalization of Steinhaus’ 3 gap theorem.

Harold Diamond (Illinois)

A survey of some tauberian theorems

A light look at inversion theorems for Laplace transforms under various hypotheses.

Xin Zhang (Illinois)

Pair correlation in Apollonian circle packings

Montgomery and Dyson’s Pair Correlation Conjecture says that the pair correlation of the non-trivial zeros of the Riemann zeta function agrees with the pair correlation of the eigenvalues of a random Hermitian matrix. Despite its huge influence, a resolution of this conjecture seems far away.  Pair correlations for some other deterministic sequences can otherwise be determined rigorously. In this talk, I will show that the limiting pair correlation of the circles from a fixed Apollonian circle packing exists. A key feature in our work, which differs from previous work in literature, is that the underlying point process is fractal in nature.  A critical tool in our analysis is an extended version of a theorem of Mohammadi-Oh on the equidistribution of expanding horospheres in the frame bundles of infinite volume hyperbolic spaces. This work is motivated by an IGL project that I mentored in Spring 2017.

Chris Linden (Illinois)

Even and Odd Minkowski Question Mark Functions

We introduce and discuss analogues of Minkowski's question mark function ?(x) related to continued fraction expansions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.

James Maynard (IAS)

Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory.  The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

Bill Banks (University of Missouri)

Non-vanishing of Dirichlet series without Euler products

This talk explores the question: To what extent does the Euler product expansion of the Riemann zeta function account for the non-vanishing of the Riemann zeta function in the half-plane {Re(s)>1}?   We exhibit a family of Dirichlet series that are closely related to the Riemann zeta function and are nonzero in {Re(s)>1}, but do not possess an Euler product.

Simon Myerson (University College London)

Real and rational systems of forms

 Consider a system f consisting of R forms of degree d with integral coefficients. We seek to estimate the number of solutions to f=0 in integers of size B or less. A classic result of Birch (1962) answers this question when the number of variables is of size at least C(d)*R^2 for some constant C(d), and the zero set f = 0 is smooth. We reduce the number of variables needed to C'(d)*R, and give an extension to systems of Diophantine inequalities |f| < 1 with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality.