  We introduce a one-parameter family of series associated to the Riemann
$\zeta$-function and prove that the values of the elements of this family at
integers are linearly independent over the rationals for almost all values of
the parameter, where almost all is with respect to any sufficiently nice
measure. We also give similar results for the Euler--Mascheroni constant, for
$\sum_{n=1}^\infty \frac{1}{n^n}$ and for $\sum_{n=1}^\infty \frac{1}{n! +1}$.
Finally, specialising the criteria used, we give some new criteria for the
irrationality of $\zeta(k)$, the Euler--Mascheroni constant and the latter two
series.
