Suppose \(a_k=\zeta (2k),\) where \(\zeta \) is the Riemann zeta function. What is the value of the sum \(\sum \limits_{k=1}^{\infty }\frac{\zeta(2k)-1}{k}\)?

**Details and assumptions**

The Riemann zeta function \(\zeta(s)\) is defined as \(\sum \limits_{n=1}^{\infty} n^{-s}.\)

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