High moments of the Estermann function

Abstract

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

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions ${\sum }_{{\chi }_1,\dots ,{\chi }_k\left(modq\right)}{\left|L\left(\frac{1}{2},{\chi }_1\right)\right|}^2\cdots {\left|L\left(\frac{1}{2},{\chi }_k\right)\right|}^2{\left|L\left(\frac{1}{2},{\chi }_1\cdots {\chi }_k\right)\right|}^2$, obtaining a power saving error term.

Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing $f_{\pm }\left(a∕q\right):={\sum }_{j=0}^r{\left(\pm 1\right)}^j{b}_j$ where $\left[0;b_0,\dots ,b_r\right]$ is the continued fraction expansion of $a∕q$ we prove that for $k\ge 2$ and $q$ primes one has ${\sum }_{a=1}^{q-1}{f}_{\pm }{\left(a∕q\right)}^k\sim 2\left(\zeta {\left(k\right)}^2∕\zeta \left(2k\right)\right)q^k$ as $q\to \infty$.