## Algebra & Number Theory

### High moments of the Estermann function

Sandro Bettin

#### Abstract

For $a∕q∈ℚ$ the Estermann function is defined as $D(s,a∕q):= ∑n≥1d(n)n−s e(naq)$ 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≤a.

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions $∑χ1,…,χk(modq)|L(12,χ1)|2⋯|L(12,χk)|2|L(12,χ1⋯χk)|2$, obtaining a power saving error term.

Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing $f±(a∕q):= ∑j=0r(±1)jbj$ where $[0;b0,…,br]$ is the continued fraction expansion of $a∕q$ we prove that for $k≥2$ and $q$ primes one has $∑a=1q−1f±(a∕q)k∼2(ζ(k)2∕ζ(2k))qk$ as $q→∞$.

#### Article information

Source
Algebra Number Theory, Volume 13, Number 2 (2019), 251-300.

Dates
Revised: 8 May 2018
Accepted: 10 August 2018
First available in Project Euclid: 26 March 2019

https://projecteuclid.org/euclid.ant/1553565643

Digital Object Identifier
doi:10.2140/ant.2019.13.251

Mathematical Reviews number (MathSciNet)
MR3927047

Zentralblatt MATH identifier
07042060

#### Citation

Bettin, Sandro. High moments of the Estermann function. Algebra Number Theory 13 (2019), no. 2, 251--300. doi:10.2140/ant.2019.13.251. https://projecteuclid.org/euclid.ant/1553565643

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