Abstract
Special values of various zeta-and L-functions in number theory and related areas at integral (or almost integral) arguments have been the major subject of research over the years.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B.C. Berndt, Character transformation formulae similar to those for the Dedekind eta-function, Proc. Sympos. Pure Math. Vol. 24 Amer. Math. Soc. Providence, 9–30, 1973.
B.C. Berndt, On Eisenstein series with characters and the values of Dirichlet L-functions, Acta Arith. 28 299–320, 1975/76.
B.C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 147–189, 1977.
B.C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York-Berlin, esp. p. 279, 1985.
B.C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York-Berlin, esp. p. 276, 1989.
S. Bochner, Some properties of modular relations, Ann. of Math. 53(2), 332–363, 1951.
S. Egami, A y-analogue of a formula of Ramanujan for (1/2), Acta Arith. 69 189–191,1995.
S. Egami, On the values of Dedekind zeta function of totally real fields at positive odd arguments, preprint.
A.O. Gel’ fond, Residues and their applications, Mir Publishers 1971, translated from the Russian.
E. Grosswald, Remarks concerning the values of the Riemann zeta function at integral, odd arguments, J. Number Theory 4 225–235, 1972.
E. Grosswald, Relations between the values at integral arguments of Dirichlet series that satisfy functional equations, Proc. Sympos. Pure Math. Vol 24 Amer. Math. Soc. Providence, 111–122, 1973.
E. Grosswald, Relations between the values of zeta and L-functions at integral arguments, Acta Arith. 24 369–378, 1973.
A.P. Guinand, Some rapidly convergent series for the Riemann s-function, Quart. J. Math. Oxford Ser. 6(2), 156–160, 1955.
G.H. Hardy, On Dirichlet’s divisor problem, Proc. London. Math. Soc. 15(2) 1–25, (p. 7), 1916; Collected Papers, Vol II, 268–292, p. 274.
E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann. 112 664–699, 1936: Mathmatische Werke, Göttingen 1959 (Nr. 33, SS. 591–626).
S. Kanemitsu, H. Kumagai and M. Yoshimoto, On rapidly convergent series expressions for zeta-and L-values, and log sine integrals, The Ramanujan J. 5 91–104, 2001.
S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, On rapidly convergent series for the Riemann zeta-values via the modular relation, preprint.
S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, On zeta-and L-function values at special rational arguments via the modular relation, to appear.
S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, On zeta-function values of number fields via the modular relation, in preparation.
K. Katayama, Ramanujan’s formulas for L-functions, J. Math. Soc. Japan 26, 234–240, 1974.
K. Katayama, Zeta-functions, Lambert series and arithmetic functions analogous to Ramanujan’s t-function. I, J. Reine Angew. Math. 268/269 251–270, 1974.
M. Katsurada, Rapidly convergent series representations for ß(2n + 1) and their x-analogue, Acta Arith. 90 79–89, 1999.
M. Katsurada, On an asymptotic formula of Ramanujan for a certain theta-type series, to appear.
M.I. Knopp, Hamburger’s theorem on (s) and the abundance principle for Dirichlet series with functional equations, Number Theory (Ed. by R.P. Bambah et al.) Hindustan Book Agency, 201–216, 2000.
A.F. Lavrik, Arithmetic equivalents to functional equations of Riemann type, Trudy Mat. Inst. Steklov 200 213–221, 1991; English translation, Proc. Steklov Inst. Math. 200 237–245, 1993.
Y. Matsuoka, On the values of the Riemann zeta function at half integers, Tokyo J. Math. 2 371–377, 1979.
Y. Matsuoka, Generalizations of Ramanujan’s formulae, Acta Arith. 4119–26,1982.
Hj. Mellin, Die Dirichletschen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht, Acta Soc. Sci. Fennicæ 31 1–48, 1902, and Acta Math. 28 37–64, 1904.
C. Nagasaka, Eichler integrals and generalized Dedekind sums, Memoirs of the Faculty of Science, Kyushu University Ser. A, 37 35–43, 1983.
T. Rivoal, La fonction zêta de Riemann prend une infinité de values irrationnelles aux entiers impairs, C.R. Acad. Sci. Paris Sér. I Math. 331 267–279, 2000.
H.M. Srivastava and H. Tsumura, New rapidly convergent series representations for (2n + 1), L(2n,x) and L(2n + 1, X) Math. Sci. Res. Hot-Line 4 17–24, 2000.
E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford UP, 1937.
M. Toyoizumi, Formulae for the values of zeta and L-functions at half integers, Tokyo J. Math. 4 193–201, 1981.
M. Toyoizumi, Ramanujan’s formulae for certain Dirichlet series, Comment. Math. Univ. St. Paul. 30 149–173, 1981, ibid. 31 87, 1982.
M. Toyoizumi, On the values of the Dedekind zeta function of an imaginary quadratic field at s = 1/3, Comment. Math. Univ. St. Paul. 31 159–161, 1982.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this paper
Cite this paper
Kanemitsu, S., Tanigawa, Y., Yoshimoto, M. (2002). On Rapidly Convergent Series for Dirichlet L-function Values Via the Modular Relation. In: Agarwal, A.K., Berndt, B.C., Krattenthaler, C.F., Mullen, G.L., Ramachandra, K., Waldschmidt, M. (eds) Number Theory and Discrete Mathematics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8223-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8223-1_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9481-4
Online ISBN: 978-3-0348-8223-1
eBook Packages: Springer Book Archive