Vol. 13, No. 2, 2019

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High moments of the Estermann function

Sandro Bettin

Vol. 13 (2019), No. 2, 251–300
Abstract

For aq the Estermann function is defined as D(s,aq) := n1d(n)ns e(na q) 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 1 a < q.

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet L-functions χ1,,χk(modq)|L(1 2,χ1)|2|L(1 2,χk)|2|L(1 2,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 k 2 and q primes one has a=1q1f±(aq)k 2(ζ(k)2ζ(2k))qk as q .

Keywords
Estermann function, Dirichlet L-functions, divisor function, continued fractions, mean values, moments
Mathematical Subject Classification 2010
Primary: 11M06
Secondary: 11A55, 11M41, 11N75
Milestones
Received: 14 February 2017
Revised: 8 May 2018
Accepted: 10 August 2018
Published: 2 March 2019
Authors
Sandro Bettin
Dipartimento di Matematica
Università di Genova
16146 Genova
Italy