Bessel series expansions of the Epstein zeta function and the functional equation

Terras, Audrey A.

doi:10.1090/S0002-9947-1973-0323735-6

Bessel series expansions of the Epstein zeta function and the functional equation

Author: Audrey A. Terras
Journal: Trans. Amer. Math. Soc. 183 (1973), 477-486
MSC: Primary 10H10
DOI: https://doi.org/10.1090/S0002-9947-1973-0323735-6
MathSciNet review: 0323735
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Abstract: For the Epstein zeta function of an n-ary positive definite quadratic form, generalizations of the Selberg-Chowla formula (for the binary case) are obtained. Further, it is shown that these formulas suffice to prove the functional equation of the Epstein zeta function by mathematical induction. Finally some generalizations of Kronecker's first limit formula are obtained.

[1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
[2] Tetsuya Asai, On a certain function analogous to 𝑙𝑜𝑔_{𝜂}(𝑧), Nagoya Math. J. 40 (1970), 193–211. MR 271038
[3] P. T. Bateman and E. Grosswald, On Epstein’s zeta function, Acta Arith. 9 (1964), 365–373. MR 179141, https://doi.org/10.4064/aa-9-4-365-373
[4] Bruce C. Berndt, Identities involving the coefficients of a class of Dirichlet series. V, VI, Trans. Amer. Math. Soc. 160 (1971), 139-156; ibid. 160 (1971), 157–167. MR 0280693, https://doi.org/10.1090/S0002-9947-1971-0280693-9
[5] Proceedings of Symposia in Pure Mathematics. Vol. IX: Algebraic groups and discontinuous subgroups, Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society held at the University of Colorado, Boulder, Colorado (July 5-August 6, vol. 1965, American Mathematical Society, Providence, R.I., 1966. MR 0202512
[6] Paul Epstein, Zur Theorie allgemeiner Zetafunktionen. I, Math. Ann. 56 (1903), 614-644.
[7] Erich Hecke, Mathematische Werke, Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen, Vandenhoeck & Ruprecht, Göttingen, 1959 (German). MR 0104550
[8] Carl S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474–523. MR 69960, https://doi.org/10.2307/1969810
[9] William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
[10] Max Koecher, Ein neuer Beweis der Kroneckerschen Grenzformel, Arch. Math. 4 (1953), 316–321 (German). MR 59303, https://doi.org/10.1007/BF01899895
[11] Atle Selberg and S. Chowla, On Epstein’s zeta-function, J. Reine Angew. Math. 227 (1967), 86–110. MR 215797, https://doi.org/10.1515/crll.1967.227.86
[12] C. L. Siegel, Lectures on quadratic forms, Notes by K. G. Ramanathan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 7, Tata Institute of Fundamental Research, Bombay, 1967. MR 0271028
[13] -, Lectures on advanced analytic number theory, Tata Inst., Bombay, 1961.
[14] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
[15] Audrey Terras, A generalization of Epstein’s zeta function, Nagoya Math. J. 42 (1971), 173–188. MR 286761