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Cotangent sums, a further generlization of Dedekind sums. (English) Zbl 0537.10005

Let \(\cot(z)=\cot \pi z\) if \(z\neq\) integer, \(\cot(z)=0\) otherwise. The cotangent sums considered here are \[ (1/c)\sum_{\nu mod c}\cot(- x+a(\nu +z)/c) \cot(-y+b(\nu +z)/c) \] where a,b,c are positive integers coprime in pairs, and x,y,z are real numbers in the interval [0,1). The author derives a three-term relation which, in the special case \(bb^{- 1}\equiv 1 (mod c)\) implies a reciprocity law. Specialization of the parameters x,y,z gives results similar to those for classical Dedekind sums and some of their relatives. In particular, B. Berndt’s modified Dedekind sums [J. Reine Angew. Math. 303/304, 332-365 (1978; Zbl 0384.10011)] are special cases of these cotangent sums. Algorithms for calculating these sums are also described.
Reviewer: T.M.Apostol

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11L99 Exponential sums and character sums
11F03 Modular and automorphic functions

Citations:

Zbl 0384.10011
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References:

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