# Number Theory

## 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.

## On the Quality of the ABC-Solutions

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.

## Torsion points and concurrent lines on Del Pezzo surfaces of degree one

The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X.

A natural question arises when studying the configuration of those curves: If a point of X is contained in “many” exceptional curves, is it torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.

## Sums of Fibonacci numbers close to a power of 2

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.

## Quadratic Twists of Modular L-functions

The behavior of quadratic twists of modular L-functions at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Our proof depends on deriving

an optimal large sieve type bound.

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

## Moments and Periods for GL(3)

In the past century, the studies of moments of L-functions have been important in number theory and are well-motivated by a variety of arithmetic applications. This talk will begin with two problems in elementary number theory, followed by a survey of techniques in the past and the present. We will slowly move towards the perspectives of period integrals which will be used to illustrate the interesting structures behind moments. In particular, we shall focus on the “Motohashi phenomena”.

## Extreme Values of the Riemann Zeta Function and Dirichlet L-functions at the Critical Points of the Zeta Function

We compute extreme values of the Riemann Zeta function at the critical points of the zeta function in the critical strip. i.e. the points where $\zeta'(s) = 0$ and $\mathfrak{R}s< 1.$. We show that the values taken by the zeta function at these points are very similar to the extreme values taken without any restrictions. We will show geometric significance of such points.

We also compute extreme values of Dirichlet L-functions at the critical points of the zeta function, to the right of $\mathfrak{R}s=1$. It shows statistical independence of L-functions and zet function in a certain way as these values are very similar to the values taken by L-functions without any restriction.

## On vertex-transitive graphs with a unique hamiltonian circle

We will discuss graphs that have a unique hamiltonian cycle and are vertex-transitive, which means there is an automorphism that takes any vertex to any other vertex. Cycles are the only examples with finitely many vertices, but the situation is more interesting for infinite graphs. (Infinite graphs do not have "hamiltonian cycles," but there are natural analogues.) The case where the graph has only finitely many ends is not difficult, but we do not know whether there are examples with infinitely many ends. This is joint work in progress with Bobby Miraftab.

## Height gaps for coefficients of D-finite power series

A power series $f(x_1,\ldots,x_m)\in \mathbb{C}[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $\mathbb{C}(x_1,\ldots,x_m)$. For the univariate series $f(x)=\sum a_nx^n$, this is equivalent to the condition that the sequence $(a_n)$ is P-recursive meaning a non-trivial linear recurrence relation of the form:

$$P_d(n)a_{n+d}+\cdots+P_0(n)a_n=0$$ where the $P_i$'s are polynomials. In this talk, we consider D-finite power series with algebraic coefficients and discuss the growth of the Weil height of these coefficients. This is from a joint work with Jason Bell and Umberto Zannier in 2019 and a more recent work in June 2022.

## Multiplicative functions in short intervals

In this talk, we are interested in a general class of multiplicative functions. For a function that belongs to this class, we will relate its “short average” to its “long average”. More precisely, we will compute the variance of such a function over short intervals by using Fourier analysis and by counting rational points on certain binary forms. The discussion is applicable to some interesting multiplicative functions such as

$$

\mu_k(n), \frac{\phi (n)}{n}, \frac{n}{\phi (n)}, \mu^2(n)\frac{\phi(n)}{n},

\sigma_\alpha (n), (-1)^{\#\left\{p: p^k | n \right\}}

$$

and many others and it provides various new results and improvements to the previous result

in the literature. This is a joint work with Mithun Kumar Das.

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