A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Eie, Minking

doi:10.1090/S0002-9947-96-01479-1

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
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by Minking Eie

Trans. Amer. Math. Soc. 348 (1996), 1117-1136

DOI: https://doi.org/10.1090/S0002-9947-96-01479-1

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Abstract:

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

References

Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125, DOI 10.1007/978-1-4612-1088-7
Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 0590943
Min King Eie, On a Dirichlet series associated with a polynomial, Proc. Amer. Math. Soc. 110 (1990), no. 3, 583–590. MR 1037206, DOI 10.1090/S0002-9939-1990-1037206-0
Laurent Denis, Lemmes de zéros et intersections, Approximations diophantiennes et nombres transcendants (Luminy, 1990) de Gruyter, Berlin, 1992, pp. 99–104 (French, with English summary). MR 1176525
Gerard van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR 930101, DOI 10.1007/978-3-642-61553-5
Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393–417. MR 427231
Hans Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman. MR 0364103, DOI 10.1007/978-3-642-80615-5
Carl Ludwig Siegel, Lectures on advanced analytic number theory, Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965. Notes by S. Raghavan. MR 0262150
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
D. Zagier, Valeurs des fonctions zêta des corps quadratiques réels aux entiers négatifs, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976) Astérisque, No. 41-42, Soc. Math. France, Paris, 1977, pp. 135–151 (French). MR 0441925