Number Theory

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.

Solaleh Bolvardizadeh (University of Lethbridge, Canada)

The quality of the triplet $(a,b,c)$, where $\gcd(a,b,c) = 1$, satisfying $a + b = c$ is defined as
$$
q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)},
$$
where $\mathrm{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture states that given $\epsilon > 0$ the number of the $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1 + \epsilon$ is finite.

In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of the $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$.

In the remaining of the talk, we prove a conjecture of Erd\"os on the solutions of the Brocard-Ramanujan equation
$$
n! + 1 = m^2
$$
by assuming an explicit version of the $ABC$-conjecture proposed by Baker.

Elchin Hasanalizade (University of Lethbridge, Canada)

The Fibonacci sequence \(F(n) : (n\geq 0) is the binary recurrence sequence defined by

$$
F(0) = F(1) = 1 \qquad \mbox{and} \\
F(n+2) = F(n+1) + F(n) \qquad \forall n \geq 0.
$$

There is a broad literature on the Diophantine equations involving the Fibonacci numbers. In this talk, we will study the Diophantine inequality

$$
\left\lvert F(n) + F(m) − 2a\right\rvert < 2a/2
$$

in positive integers n,m and a with $n \geq m$. The main tools used are lower bounds for linear forms in logarithms due to Matveev and Dujella-Petho version of the Baker-Davenport reduction method in Diophantine approximation.