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Sub-Weyl subconvexity for Dirichlet $L$ -functions to prime power moduli

Published online by Cambridge University Press:  03 November 2015

Djordje Milićević
Affiliation:
Bryn Mawr College, Department of Mathematics, 101 North Merion Avenue, Bryn Mawr, PA 19010, USA email dmilicevic@brynmawr.edu
Corresponding
E-mail address:

Abstract

We prove a subconvexity bound for the central value $L(\frac{1}{2},{\it\chi})$ of a Dirichlet $L$ -function of a character ${\it\chi}$ to a prime power modulus $q=p^{n}$ of the form $L(\frac{1}{2},{\it\chi})\ll p^{r}q^{{\it\theta}+{\it\epsilon}}$ with a fixed $r$ and ${\it\theta}\approx 0.1645<\frac{1}{6}$ , breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving $p$ -adically analytic phases, which can be naturally seen as a $p$ -adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.

Type
Research Article
Copyright
© The Author 2015 

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Sub-Weyl subconvexity for Dirichlet $L$ -functions to prime power moduli
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