Mean Values of Arithmetic Functions in Function Fields - Mathematics - EPSRC DTP funded PhD Studentship

About the award

This project is one of a number funded by the Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Partnership to commence in September 2018. This project is in direct competition with others for funding; the projects which receive the best applicants will be awarded the funding.

The studentships will provide funding for a stipend which is currently £14,553 per annum for 2017-2018. It will provide research costs and UK/EU tuition fees at Research Council UK rates for 42 months (3.5 years) for full-time students, pro rata for part-time students.

Please note that of the total number of projects within the competition, up to 15 studentships will be filled.

Location: Streatham Campus, Exeter

Project Description

In this project we will study mean values of arithmetic functions over the ring of polynomials over finite fields and over higher genus function fields. The main aim is to explore mean values of arithmetic functions over short intervals and over arithmetic progressions in the function field context.

In the past few years we have seen an explosion of activity involving the study of arithmetic functions over function fields, following the pioneer work of Keating and Rudnick.

In this project we will specifically study the mean value of the divisor functions, the Mobius function and the von Mangoldt function for higher genus function fields and also the mean value on short intervals and on arithmetic progressions. This can be seen as an extension of the work of Keating and Rudnick and of many other authors.

The main aim of this research is to develop a general theory involving mean values of arithmetic functions over function fields and combine such results with random matrix theory results and equidistribution results involving zeros of L-functions.

The student will gain a good understand of the main techniques in analytic number theory and the theory of L-function and will study in depth the distribution and statistics of zeros of L-functions.

The main techniques to be learned and used in this project are those coming from analytic number theory such as sieve methods, character sums and Tauberian theorems. But it is also important to notice that to achieve the aims of this project will be important to use techniques coming from algebra and random matrix theory.

Entry Requirements

You should have or expect to achieve at least a 2:1 Honours degree, or equivalent, in Mathematics. Experience in Number Theory and Complex Analysis is desirable.

The majority of the studentships are available for applicants who are ordinarily resident in the UK and are classed as UK/EU for tuition fee purposes. If you have not resided in the UK for at least 3 years prior to the start of the studentship, you are not eligible for a maintenance allowance so you would need an alternative source of funding for living costs. To be eligible for fees-only funding you must be ordinarily resident in a member state of the EU.

Applicants who are classed as International for tuition fee purposes are NOT eligible for funding. International students interested in studying at the University of Exeter should search our funding database for alternative options.