# Scientific

## 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.

## Filtrations, Mild groups and Arithmetic in an Equivariant context

**Oussama R. Hamza (University of Western Ontario, Canada)**

Pro-p groups arise naturally in number theory as quotients of absolute Galois groups over number fields. These groups are quite mysterious. During the 60's, Koch gave a presentation of some of these quotients. Furthermore, around the same period, Jennings, Golod, Shafarevich and Lazard introduced two integer sequences (a_n) and (c_n), closely related to a special filtration of a finitely generated pro-p group G, called the Zassenhaus filtration. These sequences give the cardinality of G, and characterize its topology. For instance, we have the well-known Gocha's alternative (Golod and Shafarevich): There exists an integer n such that a_n=0 (or c_n has a polynomial growth) if and only if G is a Lie group over p-adic fields.

In 2016, Minac, Rogelstad and Tan inferred an explicit relation between a_n and c_n. Recently (2022), considering geometrical ideas of Filip and Stix, Hamza got more precise relations in an equivariant context: when the automorphism group of G admits a subgroup of order a prime q dividing p-1.

In this talk, we present equivariant relations inferred by Hamza (2022) and give explicit examples in an arithmetical context.

## Dynamics and Wakes of a Fixed and Freely Moving Angular Particle in an Inertial Flow

We investigate the interaction between a Platonic solid and an unbounded inertial flow. For a fixed Platonic particle in the flow, we consider three different angular positions: face facing the flow, edge facing the flow, and corner facing the flow, to elucidate the effects of the particle angularity on the flow regime transitions. The impact of these angular positions, notably on drag and lift coefficients, is discussed. The particle cross-section area has a prominent influence on the drag coefficients for low Reynolds numbers, but for higher Reynolds numbers, the impacts of angular positions are more significant. As for the freely moving particle, the change in symmetry of the wake region and path instabilities are strongly related to the particle's angular position and the transverse forces. We analyze and determine the two well-known regimes transitions: the loss of symmetry of the wake and the loss of stationarity of the flow.

## Infinite Systems of Linear Equations

This video has been recovered from an archive of RealMedia files created by PIMS. The original resolution of the file was 320x200 which has been upscaled for this site.

## Bregman divergence regularization of optimal transport problems on a finite set

In optimal transport problems on a finite set, one successful approach to reducing its computational burden is the regularization by the Kullback-Leibler divergence. Then a natural question arises: Are other divergences not admissible for regularization? What kinds of properties are required for divergences? I introduce required properties for Bregman divergences and provide a non-asymptotic error estimate for the optimal transport problem regularized by such Bregman divergences. This convergence is possibly faster than exponential decay as the regularized parameter goes to zero.

This talk is based on joint work with Koya Sakakibara (Okayama U. of Science) and Keiichi Morikuni (U. of Tsukuba).

## 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

## The Emergence of Spatial Patterns for Diffusion-Coupled Compartments with Activator-Inhibitor Kinetics in 1-D and 2-D

Since Alan Turing's pioneering publication on morphogenetic pattern formation obtained with reaction-diffusion (RD) systems, it has been the prevailing belief that two-component reaction diffusion systems have to include a fast diffusing inhibiting component (inhibitor) and a much slower diffusing activating component (activator) in order to break symmetry from a uniform steady-state. This time-scale separation is often unbiological for cell signal transduction pathways.

We modify the traditional RD paradigm by considering nonlinear reaction kinetics only inside compartments with reactive boundary conditions to the extra-compartmental space that provides a two-species diffusive coupling. The construction of a nonlinear algebraic system for all existing steady-states enables us to derive a globally coupled matrix eigenvalue problem for the growth rates of eigenperturbations from the symmetric steady-state, on finite domains in 1-D and 2-D and a periodically extended version in 1-D.

We show that the membrane reaction rate ratio of inhibitor rate to activator rate is a key bifurcation parameter leading to robust symmetry-breaking of the compartments. Illustrated with Gierer-Meinhardt, FitzHugh-Nagumo and Rauch-Millonas intra-compartmental reaction kinetics, our compartmental-reaction diffusion system does not require diffusion of inhibitor and activator on vastly different time scales.

Our results elucidate a possible mechanism of the ubiquitous biological cell specialization observed in nature.