Vorrapan Chandee (Kansas State University , USA)

On Benford's law for multiplicative functions

Abstract: Benford's law is a phenomenon about the first digits of the numbers in data sets. In particular, the leading digits does not exhibit uniform distribution as might be naively expected, but rather, the digit $1$ appears the most, followed by $2, 3$, and so on until $9$. In this talk, I will discuss my recent joint work with Xiannan Li, Paul Pollack and Akash Sigha Roy on a criterion to determine whether a real multiplicative function is a Benford sequence. The criterion implies that the divisor functions and Hecke eigenvalues of newforms, such as Ramanujan tau function, are Benford. In contrast to earlier work, our approach is based on Halasz's Theorem.

Alexandre de Faveri (Institute for Advanced Study, USA)

Title
Abstract:

Florent Jouve (Université de Bordeaux, France)

Moments in the Chebotarev Density Theorem

Abstract: I will report on joint work with Régis de La Bretèche and Daniel Fiorilli in which we consider weighted moments for the distribution of Frobenius elements in conjugacy classes of Galois groups of normal number field extensions. The question is inspired by results of Hooley and recent progress due to de La Bretèche--Fiorilli concerning moments for the distribution of primes in arithmetic progressions . As in the latter case, our results are conditional on GRH and confirm that the moments considered should be Gaussian. Time permitting we will mention another notion of moments for which particular Galois group structures exclude a Gaussian behaviour.

Robert Lemke Oliver (Tufts University, USA)

Title
Abstract:

Lilian Matthiesen (KTH Royal Institute of Technology, Sweden)

Title
Abstract:

Arul Shankar (Toronto University, Canada)

Frobenius equidistribution in families of number fields

Abstract: I will describe the notion of Frobenius equidistribution (also called Sato--Tate equidistribution) in families of number fields. In the fundamental case of families of $S_n$-number fields, this equidistribution is only known in the case $n=3$, due to Davenport--Heilbronn, as well as $n=4$ and $5$, due to Bhargava. Moreover, assuming this equidistribution, Bhargava uses the Serre mass formula to develop heuristics for the asymptotics of $S_n$-number fields, when they are ordered by discriminant.
I will then discuss results in two different directions. In the first direction, we consider the following question: what do we expect to happen when we order fields by natural invariants other than the discriminant? To shed some light on this, I will describe joint work with Frank Thorne in which we give a complete answer in the case of $S_3$ cubic fields. Second, I will describe joint work with Jacob Tsimerman, in which we develop heuristics which give evidence for Frobenius equidistribution in families of all $S_n$ number fields.

Anders Södergren (Chalmers University of Technology, Sweden)

Non-vanishing at the central point of the Dedekind zeta functions of non-Galois cubic fields

Abstract: It is believed that for every $S_n$-number field, i.e. every degree $n$ extension of the rationals whose normal closure has Galois group $S_n$, the Dedekind zeta function is non-vanishing at the central point. In the case $n = 2$ Soundararajan established, in spectacular work improving on earlier work of Jutila, the non-vanishing of the Dedekind zeta function for at least 87.5% of the fields in certain families of quadratic fields. In this talk, I will present joint work with Arul Shankar and Nicolas Templier, in which we study the case $n=3$. In particular, I will discuss some of the main ideas in our proof that the Dedekind zeta functions of infinitely many $S_3$-fields have non-vanishing central value.

Nina Zubrilina (Princeton University, USA)

Root Number Correlation Bias of Fourier Coefficients of Modular Forms

Abstract: In a recent machine learning-based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the $P$-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number. In my talk, I will discuss this root number correlation bias when the average is taken over all weight 2 modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly.