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Generalizations of a cotangent sum associated to the Estermann zeta function. (English) Zbl 1346.11045
The authors study sums of the form \[ c_0\left(\frac{r}{b}\right)=-\sum\limits_{m=1}^{b-1}\frac{m}{b} \cot \left(\frac{\pi mr}{b}\right) \] where \(r,b\in \mathbb N\), \(b\geq 2\), \(1\leq r\leq b\) and \((r,b)=1\). The above sums are related to zeroes of the Estermann zeta function [T. Estermann, Proc. Lond. Math. Soc. (2) 31, 123–133 (1930; JFM 56.0174.02)]. The authors prove the existence of a unique positive measure \(\mu\) on \(\mathbb R\), with respect to which certain normalized cotangent sums are equidistributed. New asymptotic formulas for cotangent sums and their moments are obtained, as well as an estimate for the rate of growth of the moments of order \(2k\), as a function of \(k\).

MSC:
11L03 Trigonometric and exponential sums, general
11M41 Other Dirichlet series and zeta functions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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