Okamoto, Takuya Some relations among Apostol-Vu double zeta values for coordinatewise limits at non-positive integers. (English) Zbl 1269.11085 Tokyo J. Math. 34, No. 2, 353-366 (2011). Summary: We consider Apostol-Vu double zeta values for coordinatewise limits at non-positive integers, and we give some relations among Riemann’s zeta values, Euler-Zagier double zeta values and Apostol-Vu double zeta values for all coordinatewise limits at non-positive integers. Using the relations, we also give relations among multiple Bernoulli numbers. Cited in 1 Document MSC: 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11M41 Other Dirichlet series and zeta functions × Cite Format Result Cite Review PDF Full Text: DOI References: [1] S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith., 98 (2001), 107-116. · Zbl 0972.11085 · doi:10.4064/aa98-2-1 [2] S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J., 5 (2001), 327-351. · Zbl 1002.11069 · doi:10.1023/A:1013981102941 [3] T. M. Apostol and T. H. Vu, Dirichlet series related to the Riemann zeta function, J. Number Theory, 19 (1984), 85-102. · Zbl 0539.10032 · doi:10.1016/0022-314X(84)90094-5 [4] L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol, 20 (1775), 140-186, reprinted in Opera Omnia ser. I, vol. 15 , B. G. Teubner, Berlin, 1927, 217-267 [5] A. Granville, A decomposition of Riemann’s zeta-function, in: London Math. Soc. Lectuer Note Ser., 247 , Cambridge Univ. Press, 1997, 95-101. · Zbl 0907.11024 · doi:10.1017/CBO9780511666179.009 [6] M. Hoffman, Multiple harmonic series, Pscific J. Math., 152 (1992), 275-290. · Zbl 0763.11037 · doi:10.2140/pjm.1992.152.275 [7] G. Kawashima, A class of relations among multiple zeta values, J. Number Theory, 129 (2009), 755-788. · Zbl 1220.11103 · doi:10.1016/j.jnt.2008.11.002 [8] Y. Komori, An integral representation of Mordell-Tornheim double zeta function and its values at non-positive integers, Ramanujan J., 19 (2008), 163-183. · Zbl 1175.11048 · doi:10.1007/s11139-008-9130-4 [9] Y. Komori, An integral representation of multiple Hurwitz-Lerch zeta function and generalized multiple Bernoulli numbers, Quart. J. Math. (2009), 1-60. · Zbl 1270.11089 · doi:10.1093/qmath/hap004 [10] K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in: M. A. Bennett et al.(Eds), Number Theory for the Millennium II, Proc. Millennial Conference on Number Theory, A K Peters, Wellesley, 2002, pp., 417-440. · Zbl 1031.11051 [11] K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory, 101 (2003), 223-243. · Zbl 1083.11057 · doi:10.1016/S0022-314X(03)00041-6 [12] K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J., 172 (2003), 59-102. · Zbl 1060.11053 [13] K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, In: Proc. Session in Analytic Number Theory and Diophantine Equations, (eds. D. R. Heath-Brown and B. Z. Moroz), Bonner Math. Schriften, 360 , Bonn, 2003, n. 25, 17pp. · Zbl 1056.11049 [14] K. Matsumoto and H. Tsumura, On Witten multiple zeta-functions associated with semisimple Lie algebras I, Ann. Inst. Fourier (Grenoble), 56 (2006), 1457-1504 · Zbl 1168.11036 · doi:10.5802/aif.2218 [15] Y. Ohno, A genaralization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74 (1999), 39-43. · Zbl 0920.11063 · doi:10.1006/jnth.1998.2314 [16] T. Okamoto, Generalizations of Apostol-Vu and Mordell-Tornheim multiple zeta functions, Acta Arith., 140 (2009), 169-187. · Zbl 1230.11107 · doi:10.4064/aa140-2-5 [17] Y. Sasaki, Multiple zeta values for coordinatewise limits at non-positive integers, Acta Arith., 136 (2009), 299-317. · Zbl 1228.11133 · doi:10.4064/aa136-4-1 [18] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 1927. · JFM 53.0180.04 [19] D. Zagier, Values of zeta functions and their applications, In: First European Congr. Math. Vol. II, (eds. A. Joseph et al.), Progr. Math., 120 , Birkhäuser, 1994, pp. 497-512. · Zbl 0822.11001 [20] J. Q. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 128 (2000), 1275-1283. · Zbl 0949.11042 · doi:10.1090/S0002-9939-99-05398-8 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.