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

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.

### 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
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### References:

  Arfken, G., Mathematical Methods for Physicists (1970), Academic Press: Academic Press New York · Zbl 0135.42304  Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J., Evaluations of $$kk$$, Electron. J. Combin., 4(2) (1997) · Zbl 0884.40004  Borwein, J. M.; Girgensohn, R., Evaluations of triple Euler sums, Electron. J. Combin., 3 (1996) · Zbl 0884.40005  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  Euler, L., Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol., 20, 140-186 (1775)  Hoffman, M. E., Multiple harmonic series, Pacific J. Math., 152, 275-290 (1992) · Zbl 0763.11037  Kassel, C., Quantum Groups (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0808.17003  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  Macdonald, I. G., Symmetric Functions and Hall Polynomials (1995), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0487.20007  Malvenuto, C.; Reutenauer, C., Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 967-982 (1995) · Zbl 0838.05100  Markett, C., Triple sums and the Riemann zeta function, J. Number Theory, 48, 113-132 (1994) · Zbl 0810.11047  Mordell, L. J., On the evaluation of some multiple series, J. London Math. Soc. (2), 33, 368-371 (1958) · Zbl 0081.27501  Reutenauer, C., Free Lie Algebras (1993), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0798.17001  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
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