In this article, we consider various generalizations of Euler's famous relation \[ \gamma = \sum_{n\geq 2} (-1)^n \frac{\zeta(n)}{n} \] linking Euler's constant $\gamma$ to special values of the Riemann zeta function at positive integers. Among other things, we highlight the existence of a very similar relation for the Apostol-Vu harmonic zeta function which have never been noticed before.