The algebra of multiple harmonic series. (English) Zbl 0881.11067

Hoffman, Michael E.

J. Algebra 194, No. 2, 477-495 (1997).

The series in the title is an extended Riemann zeta function defined by \[ \zeta(i_1,i_2,\dots,i_k)= \sum n_1^{-i_1} n_2^{- i_2}\cdots n_k^{-i_k}, \] where the \(k\)-fold sum is over \(n_1> n_2>\cdots> n_k\geq 1\) and each \(i_r\) is a positive integer with \(i_1>1\). The sum of exponents \(i_1+ i_2+\cdots+ i_k\) is called the weight of the series. The case \(k=1\) is the Dirichlet series for the Riemann zeta function \(\zeta(i_1)\). Work by Euler, who studied the case \(k=2\) in 1775, initiated a large literature on these series and related sums. Many of the results consist of striking relations, ranging from simple properties such as \(\sum_{n=1}^\infty n^{-2} \sum_{k=1}^n k^{-1}= 2\zeta(3)\), first given by Euler and rediscovered by Ramanujan and many others, to more complicated relations such as \(\zeta(2) \zeta(2,1)= 2\zeta(2,2,1)+ \zeta(2,1,2)+ \zeta(4,1)+ \zeta(2,3)\).
This paper places these relations in a global setting by introducing an algebra that formalizes the algebraic structure arising from multiplication of these series. Among other results it is shown that there are at most a finite number of series of weight \(n\) that are irreducible, in the sense that they are not sums of rational multiples of products of series of lower weights.

Reviewer: T.M.Apostol (Pasadena)

MSC:

11M41	Other Dirichlet series and zeta functions
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
05E05	Symmetric functions and generalizations
17B01	Identities, free Lie (super)algebras
16S99	Associative rings and algebras arising under various constructions

Keywords:

algebra of multiple harmonic series; extended Riemann zeta function

References:

[1]	Arfken, G., Mathematical Methods for Physicists (1970), Academic Press: Academic Press New York · Zbl 0135.42304
[3]	Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J., Evaluations of \(kk\), Electron. J. Combin., 4(2) (1997) · Zbl 0884.40004
[4]	Borwein, J. M.; Girgensohn, R., Evaluations of triple Euler sums, Electron. J. Combin., 3 (1996) · Zbl 0884.40005
[5]	Broadhurst, D. J.; Kreimer, D., Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B, 393, 403-412 (1997) · Zbl 0946.81028
[6]	Euler, L., Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol., 20, 140-186 (1775)
[8]	Hoffman, M. E., Multiple harmonic series, Pacific J. Math., 152, 275-290 (1992) · Zbl 0763.11037
[9]	Kassel, C., Quantum Groups (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0808.17003
[10]	Le, T. Q.T.; Murakami, J., Kontsevich’s integral for the Homfly polynomial and relations between values of the multiple zeta functions, Topology Appl., 62, 193-206 (1995) · Zbl 0839.57007
[11]	Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0487.20007
[12]	Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 967-982 (1995) · Zbl 0838.05100
[13]	Markett, C., Triple sums and the Riemann zeta function, J. Number Theory, 48, 113-132 (1994) · Zbl 0810.11047
[14]	Mordell, L. J., On the evaluation of some multiple series, J. London Math. Soc. (2), 33, 368-371 (1958) · Zbl 0081.27501
[15]	Reutenauer, C., Free Lie Algebras (1993), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0798.17001
[16]	Zagier, D., Values of zeta functions and their applications, First European Congress of Mathematics (1994), Birkhauser Boston: Birkhauser Boston Cambridge, p. 497-512 · Zbl 0822.11001