Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On Mordell-Tornheim zeta values


Author: Hirofumi Tsumura
Journal: Proc. Amer. Math. Soc. 133 (2005), 2387-2393
MSC (2000): Primary 40B05; Secondary 11M06, 30B99, 33E20, 40A05
DOI: https://doi.org/10.1090/S0002-9939-05-08132-3
Published electronically: March 21, 2005
MathSciNet review: 2138881
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Mordell-Tornheim zeta value of depth $r$ can be expressed as a rational linear combination of products of the Mordell-Tornheim zeta values of lower depth than $r$ when $r$ and its weight are of different parity.


References [Enhancements On Off] (What's this?)

  • 1. M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), 275-290. MR 1141796 (92i:11089)
  • 2. J. G. Huard, K. S. Williams and Z. Nan-Yue, On Tornheim's double series, Acta Arith. 75 (1996), 105-117. MR 1379394 (97f:11073)
  • 3. K. Ihara, M. Kaneko and D. Zagier, Derivation and double shuffle relations for multiple zeta values, Preprints of the Max-Planck-Institut für Mathematik, 2004-100 (2004).
  • 4. K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in ``Number Theory for the Millennium II, Proc. of the Millennial Conference on Number Theory" M. A. Bennett et al. (eds.), A K Peters, 2002. MR 1956262 (2004a:11094)
  • 5. K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, Proceedings of the Session in Analytic Number Theory and Diophantine Equations (Bonn, January-June 2002), D. R. Heath-Brown and B. Z. Moroz (eds.), Bonner Mathematische Schriften Nr. 360, Bonn 2003, No. 25, 17 pp. MR 2075634
  • 6. L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33 (1958), 368-371. MR 0100181 (20:6615)
  • 7. M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), 245-255. MR 0797027 (87c:11091)
  • 8. L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303-314. MR 0034860 (11:654a)
  • 9. H. Tsumura, Combinatorial relations for Euler-Zagier sums, Acta Arith. 111 (2004), 27-42. MR 2038060 (2005a:11140)
  • 10. H. Tsumura, On a class of combinatorial relations for the Mordell-Tornheim triple series, preprint, submitted for publication.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 40B05, 11M06, 30B99, 33E20, 40A05

Retrieve articles in all journals with MSC (2000): 40B05, 11M06, 30B99, 33E20, 40A05


Additional Information

Hirofumi Tsumura
Affiliation: Department of Management Informatics, Tokyo Metropolitan College, Akishima, Tokyo 196-8540, Japan
Address at time of publication: Department of Mathematics, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
Email: tsumura@tmca.ac.jp, tsumura@comp.metro-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-05-08132-3
Keywords: Tornheim's double series, multiple zeta values, Riemann's zeta function, uniformly convergent series
Received by editor(s): July 7, 2003
Received by editor(s) in revised form: March 20, 2004
Published electronically: March 21, 2005
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society