Algebra & Number Theory

High moments of the Estermann function

Sandro Bettin

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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.

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11N75: Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx]

Estermann function Dirichlet L-functions divisor function continued fractions mean values moments


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

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