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.


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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