## Abstract

For $a\u2215q\in \mathbb{Q}$ the Estermann function is defined as $D\left(s,a\u2215q\right):={\sum}_{n\ge 1}d\left(n\right){n}^{-s}e\left(n\frac{a}{q}\right)$ if $\Re \left(s\right)>1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D\left(s,a\u2215q\right)$ at the central point $s=1\u22152$, when averaging over $1\le a<q$.

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions ${\sum}_{{\chi}_{1},\dots ,{\chi}_{k}\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}q\right)}{\left|L\left(\frac{1}{2},{\chi}_{1}\right)\right|}^{2}\cdots {\left|L\left(\frac{1}{2},{\chi}_{k}\right)\right|}^{2}{\left|L\left(\frac{1}{2},{\chi}_{1}\cdots {\chi}_{k}\right)\right|}^{2}$, obtaining a power saving error term.

Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing ${f}_{\pm}\left(a\u2215q\right):={\sum}_{j=0}^{r}{\left(\pm 1\right)}^{j}{b}_{j}$ where $\left[0;{b}_{0},\dots ,{b}_{r}\right]$ is the continued fraction expansion of $a\u2215q$ we prove that for $k\ge 2$ and $q$ primes one has ${\sum}_{a=1}^{q-1}{f}_{\pm}{\left(a\u2215q\right)}^{k}\sim 2\left(\zeta {\left(k\right)}^{2}\u2215\zeta \left(2k\right)\right){q}^{k}$ as $q\to \infty $.

## Citation

Sandro Bettin. "High moments of the Estermann function." Algebra Number Theory 13 (2) 251 - 300, 2019. https://doi.org/10.2140/ant.2019.13.251

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