Number Theory

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.)

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

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.

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

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.