# Number Theory

## Euler's divergent series and primes in arithmetic progressions

Euler's divergent series $\sum_{n>0} n!z^n$ which converges only for $z = 0$ becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.

During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

## A new explicit bound for the Riemann zeta function

I give a new explicit bound for the Riemann zeta function on the critical line. This is joint work with Dhir Patel and Andrew Yang. The context of this work highlights the importance of reliability and reproducibility of explicit bounds in analytic number theory.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

## A logarithmic improvement in the Bombieri-Vinogradov theorem

We improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting ($\log

x)^2$ instead of $\left(log x\right)^(5/2)$. We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar?authuser=0

## An explicit error term in the prime number theorem for large x

In 1896, the prime number theorem was established, showing that π(x) ∼ li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one can use to obtain good bounds on these error terms when x is large. Moreover, we will explore the many ways in which these bounds could be improved in the future.

## Kummer Theory for Number Fields

**Antonella Perucca (University of Luxembourg, Luxembourg)**

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.

## Least quadratic non-residue and related problems

TBA

## Zeros of linear combinations of L-functions near the critical line

In this talk, I will present a recent joint work with Yoonbok Lee, where we investigate the number of zeros of linear combinations of $L$-functions in the vicinity of the critical line. More precisely, we let $L_1, \dots, L_J$ be distinct primitive $L$-functions belonging to a large class (which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$), and $b_1, \dots, b_J$ be real numbers. Our main result is an asymptotic formula for the number of zeros of $F(\sigma+it)=\sum_{j\leq J} b_j L_j(\sigma+it)$ in the region $\sigma\geq 1/2+1/G(T)$ and $t\in [T, 2T]$, uniformly in the range $\log \log T \leq G(T)\leq (\log T)^{\nu}$, where $\nu\asymp 1/J$. This establishes a general form of a conjecture of Hejhal in this range. The strategy of the proof relies on comparing the distribution of $F(\sigma+it)$ to that of an associated probabilistic random model.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar?authuser=0

## A walk on Legendre paths

The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime p. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call "Legendre paths", which encode information about the values of the Legendre symbol. The Legendre path modulo p is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo p. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? and what proportion of the path is above the real axis? Among our results, we prove that these paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.

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## Negative moments of the Riemann zeta-function

**Alexandra Florea (University of California Irvine, USA)**

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta-function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. I will also discuss some applications to the question of obtaining cancellation of averages of the Mobius function. Joint work with H. Bui.

## The value distribution of the Hurwitz zeta function with an irrational shift

The Hurwitz zeta function $\zeta(s, \alpha)$ is a shifted integer analogue of the Riemann zeta function which shares many of its properties, but is not an ”L-function” under any reasonable definition of the word. We will first review the basics of the value distribution of the Riemann zeta function in the critical strip (moments, Bohr–Jessen theory...) and then contrast it with the value distribution of the Hurwitz zeta function.

Our focus will be on shift parameters $\alpha / \in \mathbb{Q}$, i.e., algebraic irrational or transcendental. We will present a new result (joint with Winston Heap) on moments of these objects on the critical line.