×

Symmetric Tornheim double zeta functions. (English) Zbl 1473.11170

Summary: Let \(s,t,u \in \mathbb{C}\) and \(T(s, t, u)\) be the Tornheim double zeta function. In this paper, we investigate some properties of symmetric Tornheim double zeta functions which can be regarded as a desingularization of the Tornheim double zeta function. As a corollary, we give explicit evaluation formulas or rapidly convergent series representations for \(T(s, t, u)\) in terms of series of the gamma function and the Riemann zeta function.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apostol, TM, Introduction to Analytic Number Theory (1976), Springer, New York: Undergraduate Texts in Mathematics, Springer, New York · Zbl 0335.10001 · doi:10.1007/978-1-4757-5579-4
[2] Borwein, JM, Hilbert’s inequality and Witten’s zeta-function, Am. Math. Monthly, 115, 2, 125-137 (2008) · Zbl 1148.26023 · doi:10.1080/00029890.2008.11920505
[3] Borwein, JM; Dilcher, K., Derivatives and fast evaluation of the Tornheim zeta function, Ramanujan J., 45, 2, 413-432 (2018) · Zbl 1433.11106 · doi:10.1007/s11139-017-9890-9
[4] Espinosa, O.; Moll, VH, The evaluation of Tornheim double sums. Part 1, J. Number Theory, 116, 200-229 (2006) · Zbl 1168.11033 · doi:10.1016/j.jnt.2005.04.008
[5] Furusho, H., Komori, Y., Matsumoto, K., Tsumura, H.: Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of \(p\)-adic multiple \(L\)-functions at arbitrary integers. Various aspects of multiple zeta values, pp. 27-66, RIMS Kokyuroku Bessatsu, B68, Res. Inst. Math. Sci. (RIMS), Kyoto, (2017) · Zbl 1421.11067
[6] Laurinčikas, A.; Garunkštis, R., The Lerch zeta-function (2002), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1028.11052
[7] Matsumoto, K.: On the analytic continuation of various multiple zeta-functions in: Number Theory for the Millennium II, In: Proc. of the Millennial Conference on Number Theory, M. A. Bennett et. al. (eds.), A. K. Peters, pp. 417-440 (2002) · Zbl 1031.11051
[8] Matsumoto, K.; Nakamura, T.; Ochiai, H.; Tsumura, H., On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arith., 132, 2, 99-125 (2008) · Zbl 1143.11035 · doi:10.4064/aa132-2-1
[9] Nakamura, T., A functional relation for the Tornheim double zeta function, Acta Arithmetica, 125, 3, 257-263 (2006) · Zbl 1153.11047 · doi:10.4064/aa125-3-3
[10] Onodera, K., A functional relation for Tornheim’s double zeta functions, Acta Arithmetica., 162, 4, 337-354 (2014) · Zbl 1292.11096 · doi:10.4064/aa162-4-2
[11] Srivastava, HM; Choi, J., Zeta and q-Zeta functions and associated series and integrals (2012), Amsterdam: Elsevier Inc, Amsterdam · Zbl 1239.33002
[12] Tsumura, H., On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, Math. Proc. Camb. Phil. Soc., 142, 395-405 (2007) · Zbl 1149.11044 · doi:10.1017/S0305004107000059
[13] Whittaker, E.T., Watson, G.N.: A course of modern analysis. Reprint of the fourth: edition, p. 1996. Cambridge University Press, Cambridge, Cambridge Mathematical Library (1927) · JFM 45.0433.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.