Some new proof of the sum formula and restricted sum formula

Main Article Content

ChungYie Chang


The sum formula is a basic identify of multiple zeta values that expresses a Riemann-zeta value as a homogeneous sum of multiple zeta values of given depth and weight. This formula was already known to Euler in the depth two case. Conjectured in the early 1990s, for higher depth and then proved by Granville and Zagier independently. Restricted sum formula was given in Eie \cite{2}. In this paper, we present some new proofs of those formulas.

Article Details

How to Cite
Chang, C. (2013). Some new proof of the sum formula and restricted sum formula. Tamkang Journal of Mathematics, 44(2), 123–129.


B. C. Berndt, Ramanujan's Notebooks, Part I, and II, Springer-Verlag, New York, 1985, 1989.

M. Eie, W. C. Liaw and Y. L. Ong, A restricted sum formula among multiple zeta value, J. Number Theory, 129(2009), 908--921.

M. Eie and C. S. Wei,A short proof for the sum formula and it's generalization, Arch. Math.,91(2008), 330--338.

A. Granville, A decomposition of Riemann's zeta-function, in: Analytic Number Theory, Kyoto, 1996, in: London Math. Soc. Lecture Note Ser., vol. 247,Cambridge Univ. Press, Cambridge, (1997), 95--101.

M. E. Hoffman, Multiple harmonic series, Pacific J. Math., 152 (2)(1992), 275--290.

M. E. Hoffman and C. Moen, Sums of triple harmonic series, J. Number Theory, 60(2)(1996), 329--331.

C. Markett, Triple sums and the Riemann zeta function, J. Number Theory, 48(2)(1994), 113--132.

Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74(1)(1999), 39--43.

D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., 120, Basel, Boston, Berlin: Birkhauser, 497--512, 1994.