On two recurrence formulas for two kinds of identities of Riemann zeta function (Q1898425)

Let \(\zeta (s)\) denote the Riemann zeta-function where \(s\) is any complex number. Define \[ A(n, k, l)= \sum_{a_1+ a_2+ \cdots+ a_k=n} (a_1 a_2 \cdots a_k)^l \zeta (2a_1) \zeta (2a_2) \cdots \zeta (2a_k), \] where \(n\geq k\) is a positive integer, \(a_1, a_2, a_3, \dots, a_k\) all run through positive integers satisfying \(a_1+ a_2+ \cdots+ a_k =n\). This note gives two recurrence formulas on \(A(n, k, 0)\) and \(A(n, k, 1)\), thereby solving completely the two computing problems.