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Dedekind sums and continued fractions. (English) Zbl 0785.11027

P. Barkan [C. R. Acad. Sci., Paris, Sér. A 284, 923-926 (1977; Zbl 0347.10007)] has shown that the classical Dedekind sums \(s(p,q)\) can be evaluated in terms of quotients of convergents of the continued fraction for \(p/q\) by repeated use of the reciprocity law. This is an alternate form of the evaluation of \(s(p,q)\) in terms of the remainders in the Euclidean algorithm for the \(\text{gcd}(p,q)\) given in the reviewer’s Ph. D. dissertation in 1948 and published in [Modular functions and Dirichlet series in number theory (Graduate Texts Math. 41, Springer-Verlag) (1976; Zbl 0332.10017) (2nd ed. 1990) p. 73 (Exercise 20)]. This paper gives yet another evaluation based on continued fractions and Rademacher’s three-term relation for Dedekind sums.

MSC:

11F20 Dedekind eta function, Dedekind sums
11A55 Continued fractions
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