Explicit number theory, automorphic forms and L-functions
Lead Research Organisation:
University of Bristol
Department Name: Mathematics
Abstract
The proposal concerns automorphic forms and L-functions, which are mathematical objects that encode information about sequences of interest in number theory. For instance, the so-called Dirichlet L-functions encode much of what is currently known about prime numbers. The proposed project will catalogue many other varieties of L-functions and automorphic forms, and apply the information gathered to solving number-theoretic problems. For instance, one by-product of the proposed research will be a resolution of the centuries-old Odd Goldbach Conjecture, which states that every odd integer at least 7 is the sum of three prime numbers.
Organisations
People |
ORCID iD |
| Andrew Booker (Principal Investigator) |
Publications
Adnan M
(2014)
Social dynamics of Twitter usage in London, Paris, and New York City
in First Monday
Aivazidis S
(2014)
On the distribution of the density of maximal order elements in general linear groups
in The Ramanujan Journal
Bian C
(2010)
Computing GL(3) automorphic forms
in Bulletin of the London Mathematical Society
Bian Ce
(2011)
Computing GL(3) automorphic forms
Booker A
(2016)
The Euclid-Mullin graph
in Journal of Number Theory
Booker A
(2013)
Bounds and algorithms for the -Bessel function of imaginary order
in LMS Journal of Computation and Mathematics
Booker A
(2015)
The Euclid-Mullin graph
Booker A
(2016)
Simple zeros of degree 2 $L$-functions
in Journal of the European Mathematical Society
Booker A
(2016)
A converse theorem for GL(n)
in Advances in Mathematics
Booker A
(2016)
A database of genus-2 curves over the rational numbers
in LMS Journal of Computation and Mathematics
| Description | One of the main thrusts of the fellowship proposal was the rigorous computation of automorphic forms in higher rank. Under my supervision, Ce Bian went beyond his initial groundbreaking results by developing explicit forms of the Voronoi summation formula that are suitable for numerical computation. This allowed him to compute tens of thousands of Fourier coefficients of some GL(3) automorphic forms. In another direction, my postdoc Min Lee has generalised my work with Strombergsson and Venkatesh on rigorous certification of GL(2) forms. I expect that we will be able to combine these results in the near future to obtain the first rigorous results in higher rank. The second main idea of the proposal was to develop new algorithms for computing zeros of L-functions and apply such computations to problems in explicit number theory. The first case is that of Dirichlet L-functions, which was carried out under my supervision in the thesis of David Platt, who extended the known numerical results by some six orders of magnitude. A few applications of this work (among many possible) were described in the proposal. First, Platt has used high-precision, large-scale numerical computations of the zeros of the Riemann zeta-function to give the first implementation of the Lagarias-Odlyzko algorithm for computing the prime-counting function, pi(x). This is the theoretically fastest-known algorithm, and is now the best in practice as well, i.e. Platt has used it to compute a world-record value of pi(x). Second, numerical verification of GRH can be used as a substitute for zero-free regions in some analytic problems, such as the ternary Goldbach conjecture. Indeed, with the results from Platt's thesis, Helfgott has claimed a full proof of this conjecture. |
| Exploitation Route | Certification GL(3) forms is still on-going in joint work with Min Lee. Beyond that, I have some joint work in progress with Drew Sutherland, John Voight and Dan Yasaki to explore the degree 4 L-functions associated to hyperelliptic curves of genus 2. |
| Sectors | Digital/Communication/Information Technologies (including Software),Education |