High moments of the Estermann function

Bettin, Sandro

doi:10.2140/ant.2019.13.251

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High moments of the Estermann function

Sandro Bettin

Vol. 13 (2019), No. 2, 251–300

DOI: 10.2140/ant.2019.13.251

Abstract

For $a ∕ q \in ℚ$ the Estermann function is defined as $D (s, a ∕ q) : = \sum_{n \geq 1} d (n) n^{- s} e (n \frac{a}{q})$ if $ℜ (s) > 1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D (s, a ∕ q)$ at the central point $s = 1 ∕ 2$ , when averaging over $1 \leq a < q$ .

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$ -functions $\sum_{χ_{1}, \dots, χ_{k} (mod q)} {| L (\frac{1}{2}, χ_{1}) |}^{2} \dots {| L (\frac{1}{2}, χ_{k}) |}^{2} {| L (\frac{1}{2}, χ_{1} \dots χ_{k}) |}^{2}$ , obtaining a power saving error term.

Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing $f_{\pm} (a ∕ q) : = \sum_{j = 0}^{r} {(\pm 1)}^{j} b_{j}$ where $[0; b_{0}, \dots, b_{r}]$ is the continued fraction expansion of $a ∕ q$ we prove that for $k \geq 2$ and $q$ primes one has $\sum_{a = 1}^{q - 1} f_{\pm} {(a ∕ q)}^{k} \sim 2 (ζ {(k)}^{2} ∕ ζ (2 k)) q^{k}$ as $q \to \infty$ .

Keywords

Estermann function, Dirichlet L-functions, divisor function, continued fractions, mean values, moments

Mathematical Subject Classification 2010

Primary: 11M06

Secondary: 11A55, 11M41, 11N75

Milestones

Received: 14 February 2017

Revised: 8 May 2018

Accepted: 10 August 2018

Published: 2 March 2019

Authors

Sandro Bettin
Dipartimento di Matematica Università di Genova 16146 Genova Italy

Vol. 13, No. 2, 2019