Dirichlet L-functions and their Derivatives in Function Fields
Lead Research Organisation:
UNIVERSITY OF EXETER
Department Name: Engineering Computer Science and Maths
Abstract
The proposed project is in number theory, an area of pure mathematics which is concerned with prime numbers and solutions to equations. It has long been understood in the field that there are strong analogies between number theory and the geometry of curves. As a deeper understanding of this analogy could have tremendous consequences, the search for a unified approach and the reinterpretation of geometric methods has been a source of inspiration and success for number theorists.
There is a different setting, in addition to the realm of the integer numbers, where we can study number theory, namely function fields. The main idea is that instead of studying integers and their properties we study polynomials over finite fields - a finite field is a set of numbers that behaves like the set of real numbers but contains a finite number of elements. One of the reasons to study polynomials over finite fields is that the analogies between the integers and such polynomials are striking and that by finding solutions to the function field analogue problem can lead to solutions to the classical case.
This project aims to investigate some of the analogies between numbers and polynomials by using techniques which have already been successful either for numbers or for polynomials, and by reinterpreting the geometry of curves in arithmetic terms. This will hopefully help to build a unifying theory that encapsulates these two parallel worlds. The focus of this research is to investigate a new way to tackle many simple but still unanswered questions about integers: for example, the distribution of the prime numbers, the number of divisors of different integers, and the relationship of number theory with physics. Previous researchers have studied a class of number theory questions that can be re-expressed in a form that makes them amenable to attack using techniques first developed by mathematical physicists to analyse complex quantum mechanical systems, such as large atomic nuclei and electrons subject to random forces. One of the goals of this project is to extend this connection to a much wider class of problems and to test my belief that this approach will allow answering several long-standing questions.
Possible problems include:
- Moments of L-functions in function fields and random matrix theory.
- Distribution of Prime Numbers and connections with Quantum Chaos.
- Mean values of arithmetic functions.
There is a different setting, in addition to the realm of the integer numbers, where we can study number theory, namely function fields. The main idea is that instead of studying integers and their properties we study polynomials over finite fields - a finite field is a set of numbers that behaves like the set of real numbers but contains a finite number of elements. One of the reasons to study polynomials over finite fields is that the analogies between the integers and such polynomials are striking and that by finding solutions to the function field analogue problem can lead to solutions to the classical case.
This project aims to investigate some of the analogies between numbers and polynomials by using techniques which have already been successful either for numbers or for polynomials, and by reinterpreting the geometry of curves in arithmetic terms. This will hopefully help to build a unifying theory that encapsulates these two parallel worlds. The focus of this research is to investigate a new way to tackle many simple but still unanswered questions about integers: for example, the distribution of the prime numbers, the number of divisors of different integers, and the relationship of number theory with physics. Previous researchers have studied a class of number theory questions that can be re-expressed in a form that makes them amenable to attack using techniques first developed by mathematical physicists to analyse complex quantum mechanical systems, such as large atomic nuclei and electrons subject to random forces. One of the goals of this project is to extend this connection to a much wider class of problems and to test my belief that this approach will allow answering several long-standing questions.
Possible problems include:
- Moments of L-functions in function fields and random matrix theory.
- Distribution of Prime Numbers and connections with Quantum Chaos.
- Mean values of arithmetic functions.
Organisations
People |
ORCID iD |
| Julio Andrade (Primary Supervisor) | |
| Michael Yiasemides (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509656/1 | 01/10/2016 | 30/09/2021 | |||
| 1918882 | Studentship | EP/N509656/1 | 01/10/2017 | 30/09/2020 | Michael Yiasemides |
| Description | Two main results have been proven with regards to moments of Dirichlet L-functions in function fields. For the non-specialist reader, Dirichlet L-functions are functions in mathematics that encode information about the prime numbers. The greater our understanding of such functions, the greater our understanding of primes. The "moments" of these functions are one aspect that are difficult to understand and obtain results for. One should note that prime numbers are central to cryptography. Strictly speaking, my work is with regards to prime polynomials, which are analogous to prime numbers. I also investigated another non-typical method of potentially bettering our understanding of moments of Dirichlet L-functions. |
| Exploitation Route | The other non-typical method that I investigated will likely be investigated further by myself after the end of the award. It may be a successful method, but even if not it will help others in obtaining a better understanding of moments of Dirichlet L-functions. Furthermore, I obtained some standard, yet useful, results in an area of mathematics called function fields, which are likely to be used by others working in this area. |
| Sectors | Security and Diplomacy |
| Description | Findings from this award have yet to contribute to non-academic impacts, but it is necessary to explain that there will be a long term impact and why. When Newton and Leibniz discovered calculus centuries ago, it had very few applications at the time. Nowadays, almost everything uses calculus at some point: engineering, manufacturing, economics, medicine, and more. Similarly, my work in number theory does not have an immediate impact, although in the future it will be part of a wider base of research that has impacts on the field of cryptography and security, and likely more. This is the nature of pure mathematics. |
| First Year Of Impact | 2020 |
| Sector | Security and Diplomacy |