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

Abstract

For aq the Estermann function is defined as D(s,aq):=n1d(n)ns e(naq) if (s)>1 and by meromorphic continuation otherwise. For q prime, we compute the moments of D(s,aq) at the central point s=12, when averaging over 1a<q.

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,χ1χk)|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±(aq):=j=0r(±1)jbj where [0;b0,,br] is the continued fraction expansion of aq we prove that for k2 and q primes one has a=1q1f±(aq)k2(ζ(k)2ζ(2k))qk as q.

Citation

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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

Information

Received: 14 February 2017; Revised: 8 May 2018; Accepted: 10 August 2018; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 07042060
MathSciNet: MR3927047
Digital Object Identifier: 10.2140/ant.2019.13.251

Subjects:
Primary: 11M06
Secondary: 11A55, 11M41, 11N75

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.13 • No. 2 • 2019
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