Generalized Dedekind eta-functions and generalized Dedekind sums

Berndt, Bruce C.

doi:10.1090/S0002-9947-1973-0371817-5

Generalized Dedekind eta-functions and generalized Dedekind sums
HTML articles powered by AMS MathViewer

by Bruce C. Berndt PDF

Trans. Amer. Math. Soc. 178 (1973), 495-508 Request permission

Abstract:

A transformation formula under modular substitutions is derived for a very large class of generalized Eisenstein series. The result also gives a transformation formula for generalized Dedekind eta-functions. Various types of Dedekind sums arise, and reciprocity laws are established.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147–157. MR 34781, DOI 10.1215/S0012-7094-50-01716-9
Tom M. Apostol, A short proof of Shô Iseki’s functional equation, Proc. Amer. Math. Soc. 15 (1964), 618–622. MR 164942, DOI 10.1090/S0002-9939-1964-0164942-5
L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953), 513–522. MR 56019, DOI 10.2140/pjm.1953.3.513
L. Carlitz, Dedekind sums and Lambert series, Proc. Amer. Math. Soc. 5 (1954), 580–584. MR 62764, DOI 10.1090/S0002-9939-1954-0062764-0
Ulrich Dieter, Das Verhalten der Kleinschen Funktionen $\log \sigma _{g,h}(\omega _{1}, \omega _{2})$ gegenüber Modultransformationen und verallgemeinerte Dedekindsche Summen, J. Reine Angew. Math. 201 (1959), 37–70 (German). MR 104644, DOI 10.1515/crll.1959.201.37
E. Grosswald, Dedekind-Rademacher sums and their reciprocity formula, J. Reine Angew. Math. 251 (1971), 161–173. MR 292767, DOI 10.1515/crll.1971.251.161

Bestimmung der Klassenzahl einer neuen Reihe von algebraischen Zahlkörpern

Shô Iseki, The transformation formula for the Dedekind modular function and related functional equations, Duke Math. J. 24 (1957), 653–662. MR 91301

Über die elliptischen Normalkurven der n-ten Ordnung und zugehörige Modulfunctionen der n-ten Stufe

Joseph Lehner, A partition function connected with the modulus five, Duke Math. J. 8 (1941), 631–655. MR 5523
Joseph Lewittes, Analytic continuation of Eisenstein series, Trans. Amer. Math. Soc. 171 (1972), 469–490. MR 306148, DOI 10.1090/S0002-9947-1972-0306148-1
John Livingood, A partition function with the prime modulus $P>3$, Amer. J. Math. 67 (1945), 194–208. MR 12101, DOI 10.2307/2371722
Curt Meyer, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern, Akademie-Verlag, Berlin, 1957 (German). MR 0088510
C. Meyer, Über einige Anwendungen Dedekindscher Summen, J. Reine Angew. Math. 198 (1957), 143–203 (German). MR 104643, DOI 10.1515/crll.1957.198.143
H. Rademacher, Some remarks on certain generalized Dedekind sums, Acta Arith. 9 (1964), 97–105. MR 163873, DOI 10.4064/aa-9-1-97-105
B. Schoeneberg, Zur Theorie der verallgemeinerten Dedekindschen Modulfunktionen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), 119–128 (German). MR 272714
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759