  The aim of the paper is to relate computational and arithmetic questions
about Euler's constant $\gamma$ with properties of the values of the
$q$-logarithm function, with natural choice of $q$. By these means, we
generalize a classical formula for $\gamma$ due to Ramanujan, together with
Vacca's and Gosper's series for $\gamma$, as well as deduce irrationality
criteria and tests and new asymptotic formulas for computing Euler's constant.
The main tools are Euler-type integrals and hypergeometric series.
