Mathematics

Sums of proper divisors with missing digits

Speaker: 
Kübra Benli
Date: 
Thu, Jan 25, 2024
Location: 
PIMS, University of British Columbia
Zoom
Conference: 
UBC Number Theory Seminar
Abstract: 

Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erdös, Granville, Pomerance, and Spiro conjectured that if \(\square\) is a set of integers with asymptotic density zero then the preimage set \(s^{−1}(\square)\) also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when \(\square\) is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set \(s^{-1}(\square)\). This talk is based on the joint work with Giulia Cesana, C\'{e}cile Dartyge, Charlotte Dombrowsky and Lola Thompson.

Class: 

Mean values of long Dirichlet polynomials

Speaker: 
Winston Heap
Date: 
Mon, Jan 22, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

We discuss the role of long Dirichlet polynomials in number theory. We first survey some applications of mean values of long Dirichlet polynomials over primes in the theory of the Riemann zeta function which includes central limit theorems and pair correlation of zeros. We then give some examples showing how, on assuming the Riemann Hypothesis, one can compute asymptotics for such mean values without using the Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt functions.

Class: 

Equidistribution of some families of short exponential sums

Speaker: 
Théo Untrau
Date: 
Thu, Jan 18, 2024
Location: 
PIMS, University of British Columbia
Conference: 
UBC Number Theory Seminar
Abstract: 

Exponential sums play a role in many different problems in number theory. For instance, Gauss sums are at the heart of some early proofs of the quadratic reciprocity law, while Kloosterman sums are involved in the study of modular and automorphic forms. Another example of application of exponential sums is the circle method, an analytic approach to problems involving the enumeration of integer solutions to certain equations. In many cases, obtaining upper bounds on the modulus of these sums allow us to draw conclusions, but once the modulus has been bounded, it is natural to ask the question of the distribution of exponential sums in the region of the complex plane in which they live. After a brief overview of the motivations mentioned above, I will present some results obtained with Emmanuel Kowalski on the equidistribution of exponential sums indexed by the roots modulo p of a polynomial with integer coefficients.

Class: 

Explicit bounds for $\zeta$ and a new zero free region

Speaker: 
Chiara Bellotti
Date: 
Tue, Jan 16, 2024
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region for ζ(s), showing that ζ(σ+it) has no zeros in the region σ ≥ 1-1/(53.989 (log|t|)2/3(log log|t|)1/3) for |t| ≥ 3.

Class: 

Examples of well-behaved Beurling number systems

Speaker: 
Frederik Broucke
Date: 
Thu, Dec 7, 2023
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
UBC Number Theory Seminar
Abstract: 

A Beurling number system consists a non-decreasing unbounded sequence of reals larger than 1, which are called generalized primes, and the sequence of all possible products of these generalized primes, which are called generalized integers. With both sequences one associates counting functions. Of particular interest is the case when both counting functions are close to their classical counter parts: namely when the prime-counting function is close to Li(x), and when the integer-counting function is close to ax for some positive constant a.

A Beurling number systems is well-behaved if it admits a power saving in the error terms for both these counting functions. In this talk, I will discuss some general theory of these well-behaved systems, and present some recent work about examples of such well-behaved number systems. This talk is based on joint work with Gregory Debruyne and Szilárd Révész.

Class: 

Local to global principle for higher moments of the natural density

Speaker: 
Severin Schraven
Date: 
Thu, Nov 30, 2023
Location: 
PIMS, University of British Columbia
Online
Zoom
Conference: 
UBC Number Theory Seminar
Abstract: 

In this talk I will explain how to obtain a local to global principle for expected values over free ℤ-modules of finite rank. We use the same philosophy as Ekedhal’s Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. This strategy can also be extended to higher moments and to holomorphy rings of any global function field.

These results were obtained in collaboration with A. Hsiao, J. Ma, G. Micheli, S. Tinani, V. Weger, Y.Q. Wen.

Class: 

The size function for imaginary cyclic sextic fields

Speaker: 
Ha Tran
Date: 
Tue, Nov 28, 2023
Location: 
PIMS, University of Lethbridge
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

The size function $h^0$ for a number field is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors if that field is Galois over $\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was proved for all number fields with the unit group of rank $0$ and $1$, and also for cyclic cubic fields which have unit group of rank two. In this talk, we will discuss the main idea to prove that the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two. This is joint work with Peng Tian and Amy Feaver.

Class: 

Mixing times and representation theory.

Speaker: 
Lucas Teyssier
UBC
Date: 
Wed, Dec 6, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

Consider a poker game where you have to mix the deck of cards between two turns. How (many times) should you shuffle it to prevent any cheating? In this talk we will introduce the theory of mixing times, and explain how representation theory can be used to study card mixing and diffusions on other objects.

Class: 
Subject: 

Möbius function, an identity factory with applications

Speaker: 
Sebastian Zuniga Alterman
Date: 
Mon, Dec 4, 2023
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

By using an identity relating a sum to an integral, we obtain a family of identities for the averages \(M(X)=\sum_{n\leq X} \mu(n)\) and \(m(X)=\sum_{n\leq X} \mu(n)/n\). Further, by choosing some specific families, we study two summatory functions related to the Möbius function, \(\mu(n)\), namely \(\sum_{n\leq X} \mu(n)/n^s\) and \(\sum{n\leq X} \mu(n)/n^s \log(X/n)\), where \(s\) is a complex number and \(\Re s >0\). We also explore some applications and examples when s is real. (joint work with O. Ramaré)

Class: 
Subject: 

Unbalanced Optimal Transport: Convex Relaxation and Dynamic Perspectives

Speaker: 
Giuseppe Savare
Date: 
Thu, Nov 30, 2023
Location: 
PIMS, University of Washington
Zoom
Online
Conference: 
Kantorovich Initiative Seminar
Abstract: 

I will try to present an overview of some results of unbalanced optimal transport for positive measures with different total masses, showing the crucial role of the so-called cone representation and of the corresponding homogeneous marginals. The cone perspective naturally arises in the convex-relaxation approach to optimal transport; in the more specific case of the Hellinger-Kantorovich (aka Fisher-Rao) metric, it provides a natural tool for representing solutions of the dual dynamical formulation via Hamilton-Jacobi equations, and it is very useful for studying the geodesic convexity of entropy type functionals. (In collaboration with M. Liero, A. Mielke, G. Sodini)

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Subject: 

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