Tokyo Journal of Mathematics

New Trigonometric Identities and Generalized Dedekind Sums

Shinji FUKUHARA

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Abstract

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

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

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208679

Digital Object Identifier
doi:10.3836/tjm/1244208679

Mathematical Reviews number (MathSciNet)
MR1981996

Zentralblatt MATH identifier
1048.11036

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

Citation

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


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References

  • 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.