Number Theory

Subject:

Lambert series of logarithm and a mean value theorem for $\zeta(\frac{1}{2}-it)\zeta'(\frac{1}{2}+it)$

Speaker:

Atul Dixit

Date:

Tue, Jul 26, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

An explicit transformation for the series $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny},$ Re$(y)>0$, which takes $y$ to~$\frac1y$, is obtained. This series transforms into a series containing $\psi_1(z)$, the derivative of~$R(z)$. The latter is a function studied by Christopher Deninger while obtaining an analogue of the famous Chowla--Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi_1(z)$ are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$. Our transformation readily gives the complete asymptotic expansion of $\sum_{n=1}^{\infty}d(n)\log(n)e^{-ny}$ as $y\to0$. This, in turn, gives the asymptotic expansion of $\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right)\zeta'\left(\frac{1}{2}+it\right)e^{-\delta t}\, dt$ as $\delta\to0$. This is joint work with Soumyarup Banerjee and Shivajee Gupta.

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Subject:

Negative moments of the Riemann zeta function

Speaker:

Alexandra Florea

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

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. This is joint work with H. Bui.

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869 reads

The recipe for moments of $L$-functions

Speaker:

Siegfried Baluyot

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a `recipe' that leads to detailed conjectures for the asymptotic behavior of moments of various families of $L$-functions. In this talk, we will survey recent progress towards their conjectures and explore connections with different subjects.

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1155 reads

One-level density of zeros of Dirichlet L-functions over function fields

Speaker:

Hua Lin

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

We compute the one-level density of zeros of order-$\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy.

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Subject:

Selberg's central limit theorem for quadratic Dirichlet $L$-functions over function fields

Speaker:

Allysa Lumley

Date:

Mon, Jul 25, 2022

Location:

PIMS, University of Northern British Columbia

Conference:

Moments of L-functions Workshop

Abstract:

In this talk, we will discuss the logarithm of the central value $L\left(\frac{1}{2}, \chi_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $g$ over a fixed finite field $\mathbb{F}_q$ in the limit as $g\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is asymptotically bounded above by the full Gaussian distribution of mean $\frac{1}{2}\log \deg(D)$ and variance $\log \deg(D)$, and also $\log \big|L\left(\frac{1}{2}, \chi_D\right)\big|$ is at least $94.27 \%$ Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.

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Subject:

A moment with L-functions

Speaker:

Matilde Lalín

Date:

Thu, May 12, 2022

Location:

PIMS, University of British Columbia

Online

Zoom

Conference:

PIMS Network Wide Colloquium

2022 Celebration of Women in Mathematics

Abstract:

The Riemann zeta function plays a central role in our understanding of the prime numbers. In this talk we will review some of its amazing properties as well as properties of other similar functions, the Dirichlet L-functions. We will then see how the method of moments can help us in the study of L-functions and some surprising properties of their values. This talk will be accessible to advanced undergraduate students and is part of the May12, Celebration of Women in Mathematics.

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1257 reads

Unsolved Problems in Number Theory

Speaker:

Ben Green

Date:

Wed, Sep 29, 2021

Location:

PIMS, University of Calgary

Zoom

Online

Conference:

Louise and Richard K. Guy Lecture Series

Abstract:

Richard Guy's book "Unsolved Problems in Number Theory" was one of the first mathematical books I owned. I will discuss a selection of my favorite problems from the book, together with some of the progress that has been made on them in the 30 years since I acquired my copy.

Speaker Biography

Ben Green was born and grew up in Bristol, England. He was educated at Trinity College, Cambridge and has been the Waynflete Professor of Pure Mathematics at Oxford since 2013.

About the Series

The Richard & Louise Guy Lecture Series, presented from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world, celebrates the joy of discovery and wonder in mathematics for everyone.

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2008 reads

The Life and Numbers of Richard Guy (1916 – 2020)

Speaker:

Hugh Williams

Date:

Fri, Oct 2, 2020 to Sat, Oct 3, 2020

Location:

Zoom

PIMS, University of Calgary

Conference:

Celebrating the Life of Dr. Richard Guy

Abstract:

Over fifty years ago Richard Kenneth Guy joined the then Department of Mathematics, Statistics and Computer Science at the nascent University of Calgary. Although he retired from the University in 1982, he continued, even in his last year, to come in to the University every day and work on the mathematics that he loved. In this talk I will provide a glimpse into the life and research of this most remarkable man. In doing this, I will recount several of the important events of Richard’s life and briefly discuss some of his mathematical contributions.

About Dr. Williams

: Dr. Hugh Williams is internationally recognized as an expert in computational number theory and its applications to cryptography. Shortly after obtaining his Ph.D. in 1969 from the Department of Applied Analysis and Computer Science at the University of Waterloo, he joined the newly established Department of Computer Science at the University of Manitoba, where he was promoted to the rank of Full Professor in 1979. He also served there as Associate Dean of Science for Research Development for seven years (1994-2001). He moved to the University of Calgary in 2001 to take up the iCORE Chair for Algorithmic Number Theory and Cryptography (2001-2013) and retired as Emeritus Professor of Mathematics and Statistics in 2016. Dr. Williams has authored over 150 refereed journal papers, 30 refereed conference papers and 20 books or book chapters, and from 1983-85 held a national Killam Research Fellowship. In February 2009, Dr, Williams was selected for a six year term as the inaugural Director of the Tutte Institute for Mathematics and Computing (TIMC), a highly classified research facility established by the federal government. In 2016, he was appointed Professor Emeritus in Mathematics and Statistics at the University of Calgary.

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2882 reads

The Notorious Collatz conjecture

Speaker:

Terence Tao

Date:

Fri, Oct 2, 2020

Location:

Zoom

PIMS, University of Calgary

Conference:

Louise and Richard K. Guy Lecture Series

Abstract:

Start with any natural number. If it is even, divide it by two. If instead it is odd, multiply it by three and add one. Now repeat this process indefinitely. The Collatz conjecture asserts that no matter how large an initial number one starts with, this process eventually reaches the number one (and then loops back to one indefinitely after that). This conjecture has been tested for quintillions of initial numbers, but remains unsolved in general; it is perhaps one of the simplest to state problems in all of mathematics that remains open; it is also one of the most notorious "mathematical diseases" that can lure professional and amateur mathematicians alike into devoting hours of futile effort into trying to solve the problem. While it is itself mostly a curiosity, and the full resolution still remains well out of reach of current technology, the Collatz problem is a model example of the more general concept of a dynamical system, which occurs throughout mathematics and science; and so progress on the Collatz conjecture can shed some light on the more general problem of understanding dynamical systems. In this lecture we give some of the history of the Collatz conjecture and some of its variants, and also describe some recent partial results on the problem.

About Dr. Tao:

Terence Tao was born in Adelaide, Australia in 1975. He has been a professor of mathematics at UCLA since 1999. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Fields Medal in 2006, the MacArthur Fellowship in 2007, the Waterman Award in 2008, and the Breakthrough Prize in Mathematics in 2015. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society and the National Academy of Sciences.

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