Distribution of Values of L-functions and Modular Forms
Lead Research Organisation:
King's College London
Department Name: Mathematics
Abstract
The aim of this project is to further understand the analytic behaviour of L-functions and modular forms. In particular, how the values of L-functions and modular forms are distributed and give potential applications to arithmetic or mathematical physics.
This could include exploring the distribution of zeros of Hecke cusp forms, moment estimates for families of quadratic twists of modular L-functions, random model approximations of families of L-functions, and equidistribution of mass of automorphic forms. This project could also investigate connections between L-functions and modular forms to questions in quantum chaos.
So far, the focus of the project has been on moments of the Riemann zeta function (and how these results are adapted to the case of Dirichlet L-functions) and primes and almost-primes in short intervals. The project will next explore sign changes of the Eisenstein series on the critical line.
This could include exploring the distribution of zeros of Hecke cusp forms, moment estimates for families of quadratic twists of modular L-functions, random model approximations of families of L-functions, and equidistribution of mass of automorphic forms. This project could also investigate connections between L-functions and modular forms to questions in quantum chaos.
So far, the focus of the project has been on moments of the Riemann zeta function (and how these results are adapted to the case of Dirichlet L-functions) and primes and almost-primes in short intervals. The project will next explore sign changes of the Eisenstein series on the critical line.
Organisations
People |
ORCID iD |
| Stephen James Lester (Primary Supervisor) | |
| Natalie Evans (Student) |
Publications
EVANS N
(2022)
Correlations of almost primes
in Mathematical Proceedings of the Cambridge Philosophical Society
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N50953X/1 | 30/09/2016 | 29/09/2021 | |||
| 2104688 | Studentship | EP/N50953X/1 | 30/09/2018 | 31/10/2022 | Natalie Evans |
| EP/R513064/1 | 30/09/2018 | 29/09/2023 | |||
| 2104688 | Studentship | EP/R513064/1 | 30/09/2018 | 31/10/2022 | Natalie Evans |
| EP/R513106/1 | 30/09/2018 | 29/09/2023 | |||
| 2104688 | Studentship | EP/R513106/1 | 30/09/2018 | 31/10/2022 | Natalie Evans |
| Description | The work funded through this award has resulted in a paper titled 'Correlations of almost primes', which has been published in the Mathematical Proceedings of the Cambridge Philosophical Society. The Hardy-Littlewood conjecture predicts the distribution of constellations of prime numbers among the integers. The conjecture was made in 1923, and in the case of prime pairs is known to hold on average over the difference between the two primes. In this paper, it is established that an analogue of the Hardy-Littlewood conjecture for pairs of numbers which have exactly two prime factors holds over a very short average over the difference between them. In particular, the length of the average is shorter than the average taken in the prime case. Also, the award holder worked with Hung M. Bui, Stephen Lester and Kyle Pratt on a preprint titled 'Weighted central limit theorems for central values of L-functions', which has been submitted to be considered for publication. |
| Exploitation Route | The award holder will next investigate whether these techniques used to study correlations of almost primes could be extended to other settings, such as the Gaussian integers. In particular, these techniques could be applied to the study of multiplicative functions in narrow sectors, or correlations of Gaussian almost primes. |
| Sectors | Other |