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 (Stanford University, USA)

An inequality for $GL(3)$ Fourier coefficients

Abstract: We prove a certain comparison inequality for partial sums of Fourier coefficients of Hecke-Maass cuspforms in $GL(3)$. This is a higher rank generalization of a result of Soundararajan, and has applications to distribution of mass in $GL(3)$. Joint work with Zvi Shem-Tov.

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)

Uniform exponent bounds on the number of primitive extensions of number fields

Abstract: A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach.
Here, we make progress on this weaker conjecture (which we term the "uniform exponent conjecture") in two ways. First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions. These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank. This is forthcoming work that grew out of conversations with M. Bhargava.

Lilian Matthiesen (KTH Royal Institute of Technology, Sweden)

Distributional properties of smooth numbers: Smooth numbers are orthogonal to nilsequences

Abstract: An integer is called y-smooth if all of its prime factors are of size at most y. The y-smooth numbers below x form a subset of the integers below x which is, in general, sparse but is known to enjoy good equidistribution properties in progressions and short intervals. Distributional properties of y-smooth numbers found striking applications in, for instance, integer factorisation algorithms or in work of Vaughan and Wooley on improving bounds in Waring's problem. In this talk I will discuss joint work with Mengdi Wang which considers some finer aspects of the distribution of y-smooth numbers. More precisely, we show for a very large range of the parameter y that y-smooth number are (in a certain sense) discorrelated with "nilsequences". Through work of Green, Tao and Ziegler, our result is closely related to the Diophantine problem of studying solutions to certain systems of linear equations in the set of y-smooth numbers.

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.