Number Theory

Joint value distribution of L-functions

Speaker: 
Junxian Li
Date: 
Thu, Sep 22, 2022
Location: 
PIMS, University of British Columbia
PIMS, University of Lethbridge
PIMS, University of Northern British Columbia
Zoom
Online
Conference: 
L-Functions in Analytic Number Theory Seminar
Abstract: 

It is believed that distinct primitive L-functions are “statistically independent”. The independence can be interpreted in many different ways. We are interested in the joint value distributions and their applications in moments and extreme values for distinct L-functions. We discuss some large deviation estimates in Selberg and Bombieri-Hejhal’s central limit theorem for values of several L-functions. On the critical line, values of distinct primitive L-functions behave independently in a strong sense. However, away from the critical line, values of distinct Dirichlet L-functions begin to exhibit some correlations.

This is based on joint works with Shota Inoue.

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

Class: 

Moments of the Hurwitz zeta function

Speaker: 
Anurag Sahay
Date: 
Fri, Jul 29, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\frac12$. We focus on rational $\alpha$. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters $\alpha$, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap.

Class: 

An extension of Venkatesh's converse theorem to the Selberg class

Speaker: 
Min Lee
Date: 
Fri, Jul 29, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

In his thesis, Venkatesh gave a new proof of the classical converse theorem for modular forms of level~$1$ in the context of Langlands' ``Beyond Endoscopy". We extend his approach to arbitrary levels and characters. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree~$2$ gamma factors of Selberg class type.
This is joint work with Andrew R. Booker and Michael Farmer.

Class: 

Local statistics for zeros of Artin--Schreier $L$-functions

Speaker: 
Alexei Entin
Date: 
Fri, Jul 29, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).
We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.
We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.

Class: 

Moments of $L$-functions in the world of number field counting

Speaker: 
Brandon Alberts
Date: 
Fri, Jul 29, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

We discuss some appearances of $L$-function moments in number field counting problems, with a particular focus on counting abelian extensions of number fields with restricted ramification.

Class: 

The eighth moment of the Riemann zeta function

Speaker: 
Quanli Shen
Date: 
Fri, Jul 29, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

I will talk about recent work joint with Nathan Ng and Peng-Jie Wong. We established an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture.

Class: 

Geodesic restrictions of Maass forms and moments of Hecke $L$-functions

Speaker: 
Peter Humphries
Date: 
Thu, Jul 28, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

How large are the $L^2$-restrictions of automorphic forms to closed geodesics? I will discuss how this problem can be shown to be equivalent to proving bounds for certain weighted moments of Hecke $L$-functions, and how the lattice structure of the ring of integers of real quadratic numbers fields can be exploited to obtain essentially optimal upper bounds for these weighted moments.

Class: 

Twisted first moment of $GL(3)\times GL(2)$ $L$-function

Speaker: 
Jakob Streipel
Date: 
Thu, Jul 28, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

We compute a first moment of $GL(3)\times GL(2)$ $L$-functions twisted by a $GL(2)$ Hecke eigenvalue at a prime. We talk about the ideas behind the proof, ways in which it can be generalised or extended, and obstacles for doing so in other directions. We also talk a bit about why such moments are interesting, briefly discussing some applications.

Class: 

Moments and periods for $GL(3)$

Speaker: 
Chung-Hang Kwan
Date: 
Thu, Jul 28, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

The celebrated Motohashi phenomenon concerns the duality between the fourth moment of the Riemann zeta function and the cubic moment of automorphic $L$-functions of $GL(2)$. Apart from its structural elegance, such a duality plays a very important role in various moment problems. In this talk, we will discuss the generalized Motohashi phenomena for the group $GL(3)$ through the lenses of period integrals and the method of unfolding. As a consequence, the Kuznetsov and the Voronoi formulae are not needed in our argument.

Class: 

Double square moments and bounds for resonance sums for cusp forms

Speaker: 
Praneel Samanta
Date: 
Thu, Jul 28, 2022
Location: 
PIMS, University of Northern British Columbia
Conference: 
Moments of L-functions Workshop
Abstract: 

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with
Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. These bounds reveal insights into the size and oscillation of the resonance sums and their potential resonance for $GL(4)$ forms when $k_1$ and $k_2$ are large.

Class: 

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