EPSRC Reference: 
EP/V055755/1 
Title: 
Moments of character sums and of the Riemann zeta function via multiplicative chaos 
Principal Investigator: 
Harper, Dr AJ 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
University of Warwick 
Scheme: 
New Investigator Award 
Starts: 
01 April 2022 
Ends: 
31 March 2025 
Value (£): 
174,545

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
In analytic number theory, some of our most powerful tools for studying "multiplicative" problems (e.g. problems about the distribution of prime numbers) are generating functions and characters having multiplicative properties. The most famous example of such a generating function is the Riemann zeta function, which encodes multiplicative information because it is defined by a product over primes in a certain half plane. Well known examples of multiplicative characters are the collection of Dirichlet characters mod $q$, e.g. the Legendre symbol mod $q$.
A powerful philosophy for understanding the behaviour of such functions and characters is the idea that they behave like suitable random model objects. For example, the Riemann zeta function is believed to behave in different settings like an Euler product over primes with random coefficients, or like the characteristic polynomial of a random matrix. Dirichlet characters are believed to behave like random unimodular multiplicative functions.
In recent work, I proved sharp upper and lower bounds for all the moments (that is, the power averages) of sums of random multiplicative functions, by connecting these moments with moments of short integrals of random Euler products. These short integrals are connected with the notion of multiplicative chaos from mathematical physics and probability, and can be analysed using ideas from the study of multiplicative chaos. Having completed the analysis on the random side, it is natural to want to "derandomise" and obtain the corresponding results for Dirichlet characters and for the short integrals of the Riemann zeta function.
So far, a few steps of this derandomisation have been successfully completed. I proved conjecturally sharp upper bounds for both problems (the character sum problem and the short integral problem) for low power averages. The corresponding results for higher power averages, and the corresponding lower bounds, are not yet known. On the short integral side, ArguinOuimetRadziwill have proved some related results, which however are not sharp. There has also been recent progress on lower bounds in the character sum problem, for example due to La Bret\`eche, Munsch and Tenenbaum, where again the established bounds are presumably not sharp. Very little is known about limiting distributional results, as opposed to upper and lower bounds, in either setting.
The goal of this proposal is to work out some of these missing steps of the derandomisation, with applications to the value distribution and nonvanishing of character sums and of the Riemann zeta function.

Key Findings 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Potential use in nonacademic contexts 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Impacts 
Description 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk 
Summary 

Date Materialised 


Sectors submitted by the Researcher 
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk

Project URL: 

Further Information: 

Organisation Website: 
http://www.warwick.ac.uk 