Moments of character sums and of the Riemann zeta function via multiplicative chaos
Lead Research Organisation:
University of Warwick
Department Name: Mathematics
Abstract
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, Arguin--Ouimet--Radziwill 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 non-vanishing of character sums and of the Riemann zeta function.
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, Arguin--Ouimet--Radziwill 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 non-vanishing of character sums and of the Riemann zeta function.
Organisations
People |
ORCID iD |
| Adam Harper (Principal Investigator) |
| Description | There were two main original objectives of this award, both of which have essentially been met (with the results currently being written up). The first objective was to prove that the modulus squared of the Riemann zeta function (probably the most fundamental object in analytic number theory) converges, in a certain sense, to a fundamental object from mathematical physics/probability called critical multiplicative chaos. In joint work of the PI with Saksman and Webb, we have proved this conjectured behaviour. Our proof broadly follows the strategy anticipated in the award proposal, but the final stage of the argument (proving convergence once the Riemann zeta function has essentially been replaced by a product over primes) was much more challenging than expected. To overcome this difficulty, we needed to develop a quite delicate "high moments with barriers" argument, which seems new and should be of interest also on the purely probabilistic side. The second objective was to prove lower bounds for certain averages (low moments) of Dirichlet character sums, another fundamental object from analytic number theory. In single authored work, the PI has proved such bounds on almost the full range of sum lengths, with the final range a work in progress. This argument is quite different than anticipated in the award proposal. |
| Exploitation Route | The methods developed in the work on chaos convergence for the zeta function should be applicable to the study of chaos in probability and physics, and may be taken forward by workers in those areas in situations where existing methods for proving convergence are inoperable. Having established convergence for the zeta function, there is also significant potential to find adaptations and applications, for example looking at the zeta function in shorter intervals, with possible applications to fundamental questions like its zero spacing. The study of Dirichlet characters using methods related to the award has already generated quite a lot of work by different groups of number theorists, looking at variants of the problem (e.g. character sums over short intervals), at connections with random matrix inspired conjectures, etc. Obvious further developments in this direction would include obtaining sharp bounds when averaging over real characters (as opposed to all characters), and attacking the missing range of "medium" power moments (between the second and fourth moments) where sharp bounds are not yet known. |
| Sectors | Digital/Communication/Information Technologies (including Software) |