Scientific

Zeros of linear combinations of Dirichlet L-functions on the critical line

Speaker: 
Jérémy Dousselin
Date: 
Mon, Mar 25, 2024
Location: 
PIMS, University of British Columbia
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

Fix $N\geq 1$ and let $L_1, L_2, \ldots, L_N$ be Dirichlet L-functions with distinct, primitive and even Dirichlet characters. We assume that these functions satisfy the same functional equation. Let $F(s)∶= c_1L_1(s)+c_2L_2(s)+\ldots+c_NL_N(s)$ be a linear combination of these functions ($c_j \in\mathbb{R}^*$ are distinct). $F$ is known to have two kinds of zeros: trivial ones, and non-trivial zeros which are confined in a vertical strip. We denote the number of non-trivial zeros $\rho$ with $\frac{F}(\rho)$\leq T$ by $N(T)$, and we let $N_\theta(T)$ be the number of these zeros that are on the critical line. At the end of the 90's, Selberg proved that this linear combination had a positive proportion of zeros on the critical line, by showing that $\kappa F∶=\lim \inf T (N_\theta(2T)−N_\theta(T))/(N(2T)−N(T))\geq c/N^2$ for some $c>0$. Our goal is to provide an explicit value for $c$, and also to improve the lower bound above by showing that $\kappa F \geq 2.16\times 10^{-6}/(N \log N)$, for any large enough $N$.

Class: 

The fourth moment of quadratic Dirichlet L-functions

Speaker: 
Quanli Shen
Date: 
Mon, Mar 18, 2024 to Thu, Apr 18, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

I will discuss the fourth moment of quadratic Dirichlet L-functions where we prove an asymptotic formula with four main terms unconditionally. Previously, the asymptotic formula was established with the leading main term under generalized Riemann hypothesis. This work is based on Li's recent work on the second moment of quadratic twists of modular L-functions. It is joint work with Joshua Stucky.

Class: 

Analogues of the Hilbert Irreducibility Theorem for integral points on surfaces

Speaker: 
Simone Coccia
Date: 
Thu, Mar 14, 2024
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
UBC Number Theory Seminar
Abstract: 

We will discuss conjectures and results regarding the Hilbert
Property, a generalization of Hilbert's irreducibility theorem to arbitrary
algebraic varieties. In particular, we will explain how to use conic fibrations
to prove the Hilbert Property for the integral points on certain surfaces,
such as affine cubic surfaces.

Class: 

On extremal orthogonal arrays

Speaker: 
Sho Suda
Date: 
Wed, Mar 13, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

An orthogonal array with parameters \((N,n,q,t)\) (\(OA(N,n,q,t)\) for short) is an \(N\times n\) matrix with entries from the alphabet \(\{1,2,...,q\}\) such that in any of its \(t\) columns, all possible row vectors of length \(t\) occur equally often. Rao showed the following lower bound on \(N\) for \(OA(N,n,q,2e)\):
\[ N\geq \sum_{k=0}^e \binom{n}{k}(q-1)^k, \]
and an orthogonal array is said to be complete or tight if it achieves equality in this bound. It is known by Delsarte (1973) that for complete orthogonal arrays \(OA(N,n,q,2e)\), the number of Hamming distances between distinct two rows is \(e\). One of the classical problems is to classify complete orthogonal arrays.

We call an orthogonal array \(OA(N,n,q,2e-1)\) extremal if the number of Hamming distances between distinct two rows is \(e\). In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case \(t=4\) and show how to extend it to extremal orthogonal arrays. Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.

Class: 

Interactions between topology and algebra: advances in algebraic K-theory

Speaker: 
Teena Gerhardt
Date: 
Fri, Mar 8, 2024
Location: 
PIMS, University of Regina
Zoom
Conference: 
University of Regina PIMS Distinguished Lecture
Abstract: 

The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. In this talk I will introduce algebraic K-theory and its applications, and discuss recent advances in this field.

Class: 
Subject: 

Primes in arithmetic progressions to smooth moduli

Speaker: 
Julia Stadlmann
Date: 
Mon, Mar 4, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

Class: 

L-functions in Analytic Number Theory: Biitu

Speaker: 
Bittu
Date: 
Mon, Feb 26, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

The Farey sequence FQ of order Q is an ascending sequence of fractions a/b in the unit interval (0,1] such that (a,b)=1 and 0

Class: 
Subject: 

Hilbert Class Fields and Embedding Problems

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Feb 14, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

The class number one problem is one of the central subjects in algebraic number theory that turns back to the time of Gauss. This problem has led to the classical embedding problem which asks whether or not any number field $K$ can be embedded in a finite extension $L$ with class number one. Although Golod and Shafarevich gave a counterexample for the classical embedding problem, yet one may ask about the embedding in 'Polya fields', a special generalization of class number one number fields. The latter is the 'new embedding problem' investigated by Leriche in 2014. In this talk, I briefly review some well-known results in the literature on the embedding problems. Then, I will present the 'relativized' version of the new embedding problem studied in a joint work with Ali Rajaei.

Class: 

Moments of higher derivatives related to Dirichlet L-functions

Speaker: 
Samprit Ghosh
Date: 
Wed, Feb 7, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

The distribution of values of Dirichlet L-functions \(L(s, \chi)\) for variable \(χ\) has been studied extensively and has a vast literature. Moments of higher derivatives has been studied as well, by Soundarajan, Sono, Heath-Brown etc. However, the study of the same for the logarithmic derivative \(L'(s, \chi)/ L(s, \chi)\) is much more recent and was initiated by Ihara, Murty etc. In this talk we will discuss higher derivatives of the logarithmic derivative and present some new results related to their distribution and moments at s=1.

Class: 

Consecutive sums of two squares in arithmetic progressions

Speaker: 
Vivian Kuperberg
Date: 
Fri, Feb 9, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

In 2000, Shiu proved that there are infinitely many primes whose last digit is 1 such that the next prime also ends in a 1. However, it is an open problem to show that there are infinitely many primes ending in 1 such that the next prime ends in 3. In this talk, we'll instead consider the sequence of sums of two squares in increasing order. In particular, we'll show that there are infinitely many sums of two squares ending in 1 such that the next sum of two squares ends in 3. We'll show further that all patterns of length 3 occur infinitely often: for any modulus q, every sequence (a mod q, b mod q, c mod q) appears infinitely often among consecutive sums of two squares. We'll discuss some of the proof techniques, and explain why they fail for primes. Joint work with Noam Kimmel.

Class: 

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