Department of

Mathematics


Seminar Calendar
for Number Theory Seminar events the year of Sunday, May 16, 2021.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, February 16, 2021

11:00 am in Zoom,Tuesday, February 16, 2021

Landau-Siegel zeros and central values of L-functions

Kyle Pratt (Oxford)

Abstract: Researchers have tried for many years to eliminate the possibility of Landau-Siegel zeros---certain exceptional counterexamples to the Generalized Riemann Hypothesis. Often one thinks of these zeros as being a severe nuisance, but there are many situations in which their existence allows one to prove spectacular, though illusory, results. I will review some of this history and some of these results. In the latter portion of the talk I will discuss recent work, joint with H. M. Bui and Alexandru Zaharescu, in which we show that the existence of Landau-Siegel zeros has implications for the behavior of $L$-functions at the central point.

Tuesday, March 2, 2021

11:00 am in Zoom,Tuesday, March 2, 2021

On the Liouville function at polynomial arguments

Joni Teravainen (Oxford)

Abstract: Let $\lambda$ be the Liouville function and $P(x)$ any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence $\lambda(P(n))$ changes sign infinitely often. We present a solution to this problem for new classes of polynomials $P$, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.

Tuesday, March 16, 2021

11:00 am in Zoom,Tuesday, March 16, 2021

An asymptotic local-global principle for integral Kleinian sphere packings

Edna Jones (Rutgers University)

Abstract: We will discuss an asymptotic local-global principle for certain integral Kleinian sphere packings. Examples of Kleinian sphere packings include Apollonian circle packings and Soddy sphere packings. Sometimes each sphere in a Kleinian sphere packing has a bend (1/radius) that is an integer. When all the bends are integral, which integers appear as bends? For certain Kleinian sphere packings, we expect that every sufficiently large integer locally represented as a bend of the packing is a bend of the packing. We will discuss ongoing work towards proving this for certain Kleinian sphere packings. This work uses the circle method, quadratic forms, spectral theory, and expander graphs.

For Zoom info, please email obeck@illinois.edu

Tuesday, March 23, 2021

11:00 am in Zoom (contact Olivia Beckwith at obeck@illinois.edu for details),Tuesday, March 23, 2021

Modular zeros in the character table of the symmetric group

Sarah Peluse (Princeton)

Abstract: In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity. Iíll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.

Tuesday, April 13, 2021

11:00 am in Zoom,Tuesday, April 13, 2021

Bombieri-Vinogradov type theorems for primes with a missing digit

Kunjakanan Nath (U. Montreal math)

Abstract: One of the fundamental questions in number theory is to find primes in any subset of the natural numbers. In general, it's a difficult question and leads to open problems like the twin prime conjecture, Landau's problem and many more. Recently, Maynard considered the set of natural numbers with a missing digit and showed that it contains infinitely many primes whenever the base b ≥ 10. In fact, he has established the right order of the upper and the lower bounds when the base b = 10 and an asymptotic formula whenever b is large (say 2 ◊ 10⁶). In this talk, we will consider the distribution of primes with a missing digit in arithmetic progressions for base b large enough. In particular, we will show an analog of the Bombieri-Vinogradov type theorems for primes with a missing digit for large base b. The proof relies on the circle method, which in turn is based on the Fourier structure of the digital set and the Fourier transform of primes over arithmetic progressions on an average. Finally, we will give its application to count the primes of the form p = 1 + m≤ + n≤ with a missing digit for a large base.

Tuesday, April 20, 2021

11:00 am in Zoom (contact Olivia Beckwith at obeck@illinois.edu for details),Tuesday, April 20, 2021

To Be Announced

Robert Schneider (University of Georgia)

Tuesday, April 27, 2021

11:00 am in Zoom (contact Olivia Beckwith at obeck@illinois.edu for details),Tuesday, April 27, 2021

Weighted Diophantine Approximation on Manifolds

Demi Allen (University of Bristol)

Abstract: Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. Consequently, studying the size of sets of numbers that can be approximated to within a given precision is very important. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. From a measure-theoretic point of view, this classical set of $\psi$-well-approximable numbers in Euclidean space is well understood. However, it is far less clear what happens when this set, or its various generalisations or higher dimensional analogues, is intersected with other natural sets such as curves, manifolds, or fractals. Studying such questions has become a hugely popular topic of interest in Diophantine Approximation in recent years. In this talk I will discuss recent joint work with Baowei Wang (HUST, China) concerning the particular topic of weighted Diophantine Approximation on manifolds.