Dutch Analytic Number Theory Symposium

The first edition of the Dutch Analytic Number Theory Symposium took place on 

Friday, October 18, 2024 in BBG 023 at Utrecht University.

Schedule

13:00 - 13:30   Jan-Hendrik Evertse

13:35 - 14:05   Marta Pieropan

14:10 - 14:40   Peter Koymans

14:40 - 15:00   Coffee break

15:00 - 16:00   Thomas Wright 

16:00 - 16:20   Coffee Break

16:20 - 16:50   Sebastián Carrillo Santana

16:55 - 17:25   Charlotte Dombrowsky

17:30 - 18:00   Harald Helfgott

18:30 - 20:30   Dinner*

*If you would like to join for the dinner, please e-mail L.Thompson@uu.nl

Abstracts

Sebastián Carrillo Santana (Utrecht U.)

Title: Zeros of Eisenstein Series

Abstract: For the Eisenstein series in the full modular group, Rankin and Swinnerton-Dyer showed that all the zeros in the fundamental domain lie on the unit circle. In this talk, we introduce the Eisenstein series in the congruence subgroups and study its zeros. In particular, we show that as the weight increases, the zeros in the fundamental domain get closer to the unit circle. This is joint work with Gunther Cornelissen and Berend Ringeling.

Charlotte Dombrowsky (U. Leiden)

Title: L-functions of Twists of Modular Forms

Abstract: In their paper, Males, Mono, Rolen, and Wagner develop a criterion that predicts whether the product of two L-functions associated with twists of a modular form of square-free level is zero. In this talk, we will discuss this result and compute some examples. Additionally, we will explore how this criterion can be generalized to modular forms whose level is a power of a prime. 

Jan-Hendrik Evertse (U. Leiden)

Title: Orders with few rational monogenizations

Abstract:Recall that a monogenic order is an order of the shape Z[a]={f(a): f in Z[X]}, where a is an algebraic integer. Given an order O of a number field K, the set of algebraic integers a with Z[a]=O can be divided naturally into so-called Z-equivalence classes, where two algebraic integers a,b are called Z-equivalent if a-b or a+b is a rational integer. Such a Z-equivalence class is called a monogenization of O. It is known that every order of a number field has only finitely many monogenizations and there are various results giving upper bounds for their number. 

In our talk, we generalize monogenic orders to so-called rationally monogenic orders Z_a, and consider the set of algebraic numbers a with Z_a=O for a given order O. These can be divided naturally into equivalence classes, called rationally monogenic orders. We discuss analogues of the results mentioned above for the number of rational monogenizations of an order.

Harald Helfgott (CNRS)

Title: New explicit bounds on sums of the Möbius function 

Abstract: Let \mu(n) be the Möbius function. Consider the Mertens function M(x) = \sum_{n\leq x} \mu(n), and, more generally, sums of the Möbius function with or without weight: m(x) = \sum_n \mu(n)/n, \check{m}(x) = \sum_n \log(x/n) \mu(n)/n, and so forth.

Up to now, all explicit bounds on these sums relied on an inductive process starting with elementary bounds, with the iterative step using analytic estimates on the Chebyshev function psi(x); estimating sums of mu(n) in a direct analytic way similar to that used for estimating psi(x) seemed impracticable. 

We will discuss how to get bounds on these sums through (non-obvious) analytic means. Our methods rely on L^2-bounds on Dirichlet series on the line Re s = 1. Our bounds are in all cases stronger than those in the literature, despite frequent recent progress on the matter. This is joint work with Andrés Chirre. 

Peter Koymans (Utrecht U.)

Title: Hilbert 10 via additive combinatorics

Abstract: Let R be an infinite ring that is finitely generated over Z. Hilbert’s 10th problem for R asks: does there exist an algorithm that given as input a polynomial f in R[X_1, …, X_n] outputs yes if f has a zero in R and no otherwise. Mazur and Rubin proved that Hilbert’s 10th problem is undecidable assuming finiteness of Sha. I will outline how to prove the same result without assuming finiteness of Sha. The key new input is a recent version of the Green—Tao theorem for number fields. This is joint work with Carlo Pagano.

Marta Pieropan (Utrecht U.)

Title: On the hyperbola method

Abstract: In joint work with Damaris Schindler we develop a new version of the hyperbola method for counting rational points of bounded height that generalizes the work of Blomer and Brüdern for products of projective spaces. The hyperbola method transforms a counting problem into an optimization problem on certain polytopes. I will present our results and illustrate the method in examples.

Thomas Wright (Wofford C.)

Title: Siegel zeroes and prime distributions

Abstract: One of the most famous conjectures in number theory is the Generalized Riemann Hypothesis (GRH), which states that the (non-trivial) zeroes of the Dirichlet L-function must all have real part equal to 1/2.  In this talk, we talk about some of the surprising results that could be proven if GRH were false.