  We address the problem of finding out the values of the Hurwitz zeta function
at the positive integers $k$, $\zeta(k,b)$, by working out their real and
imaginary parts separately and then combining them. A few different formulae
for the Hurwitz zeta function are known from the literature, but they are very
general and usually hold for $\Re{(k)}>1$. The advantage of formulae that only
hold at the positive integers is the fact that they are usually simpler and
easier to work with. We also obtain an analytic continuation for the generating
function of $\zeta(k,b)$ as $\sum_{k\ge 2}x^k(\zeta(k,b)-1/b^k)$, valid for
complex $x$ and $b$, where term $1/b^k$ was subtracted for convenience.
