High moments of the Estermann function

Abstract

For $a/q\in \mathbb{Q}$ the Estermann function is defined as $D(s,a/q):={\sum }_{n\ge 1}d(n)n^{-s}e(n\frac{a}{q})$ if $\Re (s)>1$ 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 $1\le a<q$. As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions ${\sum }_{{\chi }_1,\dots ,{\chi }_k\mathrm{mod}q}|L(\frac{1}{2},{\chi }_1){|}^2\cdots |L(\frac{1}{2},{\chi }_k){|}^2|L(\frac{1}{2},{\chi }_1\cdots {\chi }_k){|}^2$, 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 $k\ge 2$ and $q$ primes one has ${\sum }_{a=1}^{q-1}{f}_{\pm }(a/q{)}^k\sim 2\frac{\zeta (k{)}^2}{\zeta (2k)}{q}^k$ as $q\to \infty$.

57 pages

References in corpus (4)

• The second moment of twisted modular L-functions
• On moments of twisted $L$-functions
• On the distribution of a cotangent sum
• On the reciprocity law for the twisted second moment of Dirichlet $L$-functions

Cited by in corpus (3)

• Central values of additive twists of cuspidal $L$-functions
• Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions III: The Twisted Case
• Wide moments of $L$-functions I: Twists by class group characters of imaginary quadratic fields