Hybrid Euler-Hadamard Formula in Function Fields - Mathematics - EPSRC DTP funded PhD Studentship Ref: 2879

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.

Supervisors
Dr Julio Bueno De Andrade
Professor Andreas Langer

Location 
Streatham Campus, Exeter

Project Description

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.

The aim of this project is to understand the connections between prime numbers and zeros of L-functions by means of the Hybrid Euler-Hadarmard product formula.

The Euler-Hadamard formula is an approximation to L-functions given as a product over its nontrivial zeros multiplied by a product over the primes. The hybrid formula have been previously studied by several mathematicians and it has proved important in the study of moments of L-functions which is related to an important open problem in number theory known as Lindelöf Hypothesis. 

The main aim of the project is to extend the hybrid formula for function fields and to investigate moments of L-functions in this setting. The advantage here is that the hybrid formula will not be an approximation to L-functions but will be an exact formula since in this case the L-functions are polynomials.

Due to particular characteristics of the L-functions in function fields we expect to establish stronger results and go beyond what is known in the number field case. 

Of particular interest is the Splitting Conjecture which we intend to prove a version of it in this project. The Splitting Conjecture, roughly speaking, is the statement that the moments of L-functions are equal the moments of the products of the formulas given in the Hybrid Formula. This is a very bold conjecture known to be true only for the first few moments in the classical case. In this project we will explore this conjecture in the function field setting and by using techniques coming from other areas as such random matrix theory, quantum chaos and algebraic geometry we expect to establish a version of it in the function field setting.

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.  For information on EPSRC residency criteria click here.

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.

Summary

Application deadline:10th January 2018
Value:3.5 year studentship: UK/EU tuition fees and an annual maintenance allowance at current Research Council rate. Current rate of £14,553 per year.
Duration of award:per year
Contact: Doctoral Collegepgrenquiries@exeter.ac.uk

How to apply

You will be required to upload the following documents:
•       CV
•       Letter of application outlining your academic interests, prior research experience and reasons for wishing to
        undertake the project.
•       Transcript(s) giving full details of subjects studied and grades/marks obtained.  This should be an interim
        transcript if you are still studying.
•       If you are not a national of a majority English-speaking country you will need to submit evidence of your current
        proficiency in English.  For further details of the University’s English language requirements please see
        http://www.exeter.ac.uk/postgraduate/apply/english/.

The closing date for applications is midnight (GMT) on Wednesday 10 January 2018.  Interviews will be held at the University of Exeter in late February 2018.

If you have any general enquiries about the application process please email: pgrenquiries@exeter.ac.uk.
Project-specific queries should be directed to the supervisor.

During the application process, the University may need to make certain disclosures of your personal data to third parties to be able to administer your application, carry out interviews and select candidates.  These are not limited to, but may include disclosures to:

• the selection panel and/or management board or equivalent of the relevant programme, which is likely to include staff from one or more other HEIs;
• administrative staff at one or more other HEIs participating in the relevant programme.

Such disclosures will always be kept to the minimum amount of personal data required for the specific purpose. Your sensitive personal data (relating to disability and race/ethnicity) will not be disclosed without your explicit consent.