Illinois Number Theory Seminar 2021-2022

Lilu Zhao

The sums of unlike powers

Abstract:In this talk, we consider the problem of representations of large integers n as sums of unlike powers

We shall explain how to solve the above equation in s=13 variables. The proof involves some important methods in analytic number theory such as the circle method and the large sieve inequality. This talk is based on a joint work with J. Liu.

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Yujiao Jiang

Cancellation in additively twisted sums on GL(m)

Abstract: In this talk, we shall introduce our work concerning cancellation in additively twisted sums for any GL(m) automorphic L-function, which is uniformly in the twisted parameter. The proof relies on the generalized Bourgain-Sarnak-Ziegler criterion, the method of Montgomery and Vaughan, sieve method, and analytic theory of automorphic L-functions. This is joint work with Guangshi Lü and Zhiwei Wang.

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Oleksiy Klurman

Title: On the zeros of Fekete polynomials.

Abstract: Fekete polynomials are certain Littlewood type polynomials whose coefficients are the values of the Legendre symbol. These polynomials were considered by Fekete in an attempt to understand real zeros of Dirichlet L-functions in the critical strip. Since then, their zeros and value distribution have been a subject of an extensive study. The goal of the talk is to present some recent results concerning real zeros of Fekete polynomials.  We shall also discuss applications to sign changes of quadratic character sums and zeros of theta functions. This is a joint work with Y. Lamzouri and M. Munsch.

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Lillian Pierce

On Burgess bounds and superorthogonality

Abstract: The Burgess bound is a well-known upper bound for short multiplicative character sums, with a curious proof. It implies for example a subconvexity bound for Dirichlet L-functions. In this talk we will present two types of new work on Burgess bounds. First, we will describe new Burgess bounds in multi-dimensional settings. Second, we will present a new perspective on Burgess’s method of proof. Indeed, in order to try to improve a method, it makes sense to understand the bigger “proofscape” in which a method fits. The Burgess method hasn’t seemed to fit well into a bigger proofscape. We will show that it can be regarded as an application of “superorthogonality.” This perspective turns out to unify topics ranging across harmonic analysis and number theory. In this accessible talk, we will survey these connections, with a focus on the number-theoretic side.

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Josh Stucky

The Sixth Moment of Automorphic L-Functions

Abstract: Moments of L-functions are among the central objects of study in modern analytic number theory. In this talk I will discuss my recent results concerning the sixth moment of a family of GL(2) automorphic L-functions. Before discussing my results, I will introduce moments and families of L-functions in some generality, as well as provide some background on the specific family of L-functions I study in my paper. In particular, I will discuss previous results as well as some of the tools used to study this family. As such, the talk should be interesting and accessible to a wide audience including both experts and graduate students.

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Alex Dunn

 Bias in cubic Gauss sums: Patterson's conjecture

Abstract:

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (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 first observed by Kummer in 1846.  There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that 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.  This estimate relies on the Generalised Riemann Hypothesis and is one of the fundamental reasons why our result is conditional.

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Jori Merikoski

The polynomials and also capture their primes.

Abstract: We show that there are infinitely many primes of the form X²+(Y²+1)² and X² + (Y³+Z³)². Our work builds on the famous Friedlander-Iwaniec result on primes of the form X²+Y⁴. More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form. For the argument we need to estimate Type II sums, which is achieved by an application of the Weil bound, both for point-counting and for exponential sums over curves. The type II information we get is too narrow for an asymptotic formula, but we can apply Harman's sieve method to establish a lower bound of the correct order of magnitude for the number of primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2.

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Junxian Li

Hardy-Littlewood problems with almost primes

Abstract: The Hardy-Littlewood problem asks for the number of representations of an integer as the sum of a prime and two squares. We consider the Hardy-Littlewood problem where the two squares are restricted to almost primes. This leads to the study of primes in arithmetic progressions to large moduli and automorphic analogue of the Titchmarsh divisor problems. We also consider the number of representations of an integer as the sum of a smooth number and two almost prime squares. This is based on joint work with Assing-Blomer and Blomer-Rydin Myerson.

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Chris Lutsko

A spectral theory approach to the Apollonian counting problem

Abstract: A classic problem in number theory and group theory is to count the number of points in a group orbit satisfying a particular cut-off. In 1982, Lax and Phillips used spectral theory to obtain accurate counts of such orbits in a particular context. In 2009 this method was generalized by Kontorovich to a wider class of counting problems, however this included a restriction on the groups considered. In this talk I will present this spectral approach to counting, how to remove the restriction, and how to generalize these problems. In doing so we will see how to apply these methods to the Apollonian counting problem. This is joint work in progress with Alex Kontorovich.

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Binrong Huang

On the Rankin-Selberg problem

Abstract: In this talk, I will introduce a method to solve the Rankin-Selberg problem on the second moment of Fourier coefficients of a GL(2) Hecke eigenform. This improves the classical result 3/5 of Rankin and Selberg (in 1939/1940). If time permit, a sketch of proof and some other results will also be presented.

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Vivian Kuperberg

Odd moments in the distribution of primes

Abstract: In 2004, Montgomery and Soundararajan showed (conditionally) that the distribution of the number of primes in appropriately sized intervals is approximately Gaussian and has a somewhat smaller variance than you might expect from modeling the primes as a purely random sequence. Their work depends on evaluating sums of certain arithmetic constants that generalize the twin prime constant, known as singular series. In particular, these sums exhibit square-root cancellation in each term if they have an even number of terms, but if they have an odd number of terms, there should be slightly more than square-root cancellation. I will discuss sums of singular series with an odd number of terms, including tighter bounds for small cases and the function field analog. I will also explain how this problem is connected to a simple problem about adding fractions.

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Jiuya Wang

The sharp Erdos-Turan inequality

Abstract: Erdos and Turan prove a classical inequality for complex polynomials, which says if the polynomial attains small value on the unit circle after normalization, then all zeros will cluster around the unit circle and moreover become equidistributed in angles. The optimal constant remained the only component that is not sharp for this inequality. We will explain how tools in potential theory and energy minimization enter this question, and how they help us in characterizing the extremal distribution of zeros and proving the optimal constant. This is a joint work with Ruiwen Shu. 

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Robert Dicks

Congruences for r-colored partitions

Abstract: Let . For the partition function p(n) and  , Atkin found a number of examples of primes Q >= 5 such that there exist congruences of the form    Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$.  In this paper, for a wide range of , we prove congruences of the form  for infinitely many primes Q.  For a positive integer r, let be the r-colored partition function.   Our methods yield similar congruences for .  In particular, if r is an odd positive integer such that  for which and , then we show that there are infinitely many congruences of the form . Our methods involve the theory of modular Galois representations.

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Nick Andersen

TBA

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Niclas Technau

Randomness of sequences

Abstract: The first part of this talk surveys different notions of randomness of one-dimensional sequences. A focal point will be on what is not known about local statistics which go beyond uniform distribution modulo one. The second part is reporting on joint work with Christopher Lutsko and Athanasios Sourmelidis. By employing Fourier analytic tools, this work provides a better understanding of how random slowly growing monomial sequences are - showing that their correlations are Poissonian.