The algebra of multiple harmonic series (Q1368042)

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.