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SHORT CHARACTER SUMS AND THEIR APPLICATIONS

Part of: Zeta and $L$-functions: analytic theory Exponential sums and character sums Computational number theory

Published online by Cambridge University Press:  11 July 2023

FORREST FRANCIS Open the ORCID record for FORREST FRANCIS [Opens in a new window]
FORREST FRANCIS*
Affiliation:
School of Physical, Environmental and Mathematical Sciences, University of New South Wales, Canberra ACT 2610, Australia
*
e-mail: forrestj7@gmail.com
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Abstract

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Keywords

character sumsBurgess’ boundPólya–Vinogradov inequalityL-functionexplicit computation

MSC classification

Primary: 11L40: Estimates on character sums
Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11Y35: Analytic computations
Type
PhD Abstract
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 339 - 340
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

In analytic number theory, the most natural generalisations of the famous Riemann zeta function are the Dirichlet L-functions. Each Dirichlet L-function is attached to a q-periodic arithmetic function for some natural number q, known as a Dirichlet character modulo q. Dirichlet characters and L-functions encode with them information about the primes, especially in reference to their remainders modulo q. For this reason, number theorists are often interested in bounds on short sums of a Dirichlet character over the integers. For one, such sums appear as an intermediate step in partial summation bounds for Dirichlet L-functions. However, short character sum estimates may also be used directly to tackle other number theoretic problems.

In this thesis, we wish to examine the application of character sum estimates in some specific settings. There are three main estimates in which we are interested: the trivial bound, Burgess’ bound and the Pólya–Vinogradov inequality. Of these, we will focus primarily on Burgess’ bound. The main result of this thesis will be the computation of versions of Burgess’ bound with explicit constants for a variety of parameters, improving upon the work of Treviño [Reference Treviño5].

The interplay between Burgess’ bound and the Pólya–Vinogradov inequality is vital in this explicit setting, and we will dedicate a portion of this thesis to investigating these interactions. This makes precise the work of Fromm and Goldmakher [Reference Fromm and Goldmakher4], who demonstrated a counter-intuitive influence the Pólya–Vinogradov inequality has on Burgess’ bound.

Once we have established improvements to Burgess’ bound in the explicit setting, we will ‘test’ these improvements by tackling several applications where Burgess’ bound has been used previously. Primary among these is an explicit bound for L-functions across the critical strip. We also include applications to norm-Euclidean cyclic fields and least kth power nonresidues.

Some of this research has been published in [Reference Bordignon and Francis1–Reference Francis3].

Footnotes

Thesis submitted to the University of New South Wales in November 2022; degree approved on 2 March 2023; supervisor Timothy Trudgian.

References

Bordignon, M. and Francis, F., ‘Explicit improvements to the Burgess bound via Pólya–Vinogradov’, Preprint, 2020, arXiv:2007.08119.Google Scholar
Francis, F., ‘An investigation into several explicit versions of Burgess’ bound’, J. Number Theory 228 (2021), 87–101.10.1016/j.jnt.2021.03.018CrossRefGoogle Scholar
Francis, F., ‘Explicit subconvexity estimates for Dirichlet $L$ -functions’, Preprint, 2022, arXiv:2206.11112.Google Scholar
Fromm, E. and Goldmakher, L., ‘Improving the Burgess bound via Pólya–Vinogradov’, Proc. Amer. Math. Soc. 147(2) (2019), 461–466.10.1090/proc/14171CrossRefGoogle Scholar
Treviño, E., ‘The Burgess inequality and the least $k$ th power non-residue’, Int. J. Number Theory 11(5) (2015), 1653–1678.10.1142/S1793042115400163CrossRefGoogle Scholar