2019 NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Welcome Assistant Vice Chancellor for Research

Introduction to the rest of lectures + value distribution of L-functions away from critical line.

Coffee and light refreshments will be available in the gallery daily.

9:00- 9:30 am

10:30 - 11:00 am

2:00 - 2:30 pm

Selberg's central limit theorem and analogues in families of L-functions (typical size of values on critical line).

Updated time

Abstract: Researchers have tried for many years to eliminate the possibility of LandauSiegel 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 Dirichlet L-functions at the central point.

This talk will present an effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As an application, we obtain nontrivial average upper bounds on ℓ-torsion in the class groups of the families of fields.

Continuation of Selberg's central limit theorem and analogues in families of L-functions (typical size of values on critical line).

Larger values of L-functions on critical line -- moments, conjectures.

Abstract: I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula which relies on obtaining cancellation in averages of cubic Gauss sums over functions fields. I will also talk about the corresponding non-Kummer case when q=2 (mod 3) and I will explain why this setting is somewhat easier to handle than the Kummer case, which allows us to prove some better results.

Abstract: Moments of L-functions on the critical line (Re(s) = 1/2) have been extensively studied due to numerous applications, for example, bounds for L-functions, information on zeros of L-functions, and connections to the generalized Riemann hypothesis. However, the current understanding of higher moments is very limited. In this talk, I will give an overview how we can achieve asymptotic and bounds for higher moments by enlarging the size of various families of L-functions and show some techniques that are involved.

Progress towards moment conjectures -- upper and lower bounds.

Continuation of Progress towards moment conjectures -- upper and lower bounds.

Extreme values of L-functions.

Continuation of Extreme values of L-functions.

Updated schedule

Abstract: I will review an old trick, and relate this to some modern results involving estimates for L-functions at the edge of the critical strip. These will include a good bound for automorphic L-functions and Rankin-Selberg L-functions as well as estimates for primes which split completely in a number field.

Abstract: In the 1960's, Burgess proved a subconvexity bound for Dirichlet L-functions. However, the quality of this bound was not as strong, in terms of the conductor, as the classical Weyl bound for the Riemann zeta function. In a major breakthrough, Conrey and Iwaniec established the Weyl bound for quadratic Dirichlet L-functions. I will discuss recent work with Ian Petrow that generalizes the Conrey-Iwaniec bound for more general characters, in particular arbitrary characters of prime modulus.

Fyodorov--Keating conjectures, connections with random multiplicative functions.

Continuation of Fyodorov--Keating conjectures, connections with random multiplicative functions.

Abstract: Consider the number of squarefree integers in a randomly 
chosen short interval. In this talk we will discuss a method for 
computing the variance of such a count. The estimate we arrive at 
improves an old result of R.R. Hall and confirms a conjecture of Keating 
and Rudnick in a restricted range. This is joint work with Ofir 
Gorodetsky, Bingrong Huang, and Maksym Radziwill.