Sums of triple harmonic series (Q1126378)

For positive integers \(a,b,c\) with \(a\geq 2\), let \(A(a,b,c)=\sum i^{-a} j^{-b} k^{-c}\) extended over all integers \(i>j>k\geq 1\). The authors prove that \(\sum A(a,b,c)=\zeta(n)\) for all integers \(n\geq 4\), where \(\zeta(n)\) is the Riemann zeta-function and the sum is extended over all \(a,b,c\) whose sum is \(n\).