Tokyo Journal of Mathematics

New Trigonometric Identities and Generalized Dedekind Sums


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We obtain new trigonometric identities. We show that the coefficients of Laurent expansions of the identities give rise to the relation between special values of Hurwitz zeta function and Bernoulli numbers. Then we look into in detail the parameterized cotangent sums appearing in the identities.

Article information

Tokyo J. Math., Volume 26, Number 1 (2003), 1-14.

First available in Project Euclid: 5 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F20: Dedekind eta function, Dedekind sums
Secondary: 11L03: Trigonometric and exponential sums, general 11M35: Hurwitz and Lerch zeta functions


FUKUHARA, Shinji. New Trigonometric Identities and Generalized Dedekind Sums. Tokyo J. Math. 26 (2003), no. 1, 1--14. doi:10.3836/tjm/1244208679.

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  • T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J., 17 (1950), 147–157.
  • T. M. Apostol, Theorems on generalized Dedekind sums, Pacific J. Math., 2 (1952), 1–9.
  • B. C. Berndt, Reciprocity theorems for Dedekind sums and generalizations, Advances in Math., 23 (1977), 285–316.
  • U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory, 18 (1984), 289–305.
  • S. Fukuhara, Modular forms, generalized Dedekind symbols and period polynomials, Math. Ann., 310 (1998), 83–101.
  • S. Fukuhara, Dedekind symbols associated with J-forms and their reciprocity law, J. Number Theory, 98 (2003), 236–253.
  • S. Fukumoto, M. Furuta and M. Ue, $W$-invariants and Neumann-Siebenmann invariants for Seifert homology $3$-spheres, Topology Appl., 116 (2001), 333–369.
  • F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory. Berkeley: Publish or Perish, 1974.
  • T. Kawasaki, The index of elliptic operators over $V$-manifold, Nagoya Math. J., 84 (1981), 135–157.
  • H. Rademacher and E. Grosswald, Dedekind sums (Carus Math. Mono. No. 16). Math. Assoc. Amer., (1972).
  • R. Sczech, Dedekind sums and power residue symbols, Compositio Math., 59 (1986), 89–112.
  • D. Zagier, Higher dimensional Dedekind sums, Math. Ann., 202 (1973), 149–172.