  We apply the Euler--Maclaurin formula to find the asymptotic expansion of the
sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p
/(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q
\in\N$). The expressions often simplify considerably and the coefficients are
recognizable constants. The constant terms of the asymptotics are either
$\zeta^{(p)}(\pm q)$ (first two sums), 0 (third sum) or yield novel
mathematical constants (fourth sum). This allows numerical computation of
$\zeta^{(p)}(\pm q)$ faster than any current software. One of the constants
also appears in the expansion of the function $\sum_{n\geq 2} (n\log n)^{-s}$
around the singularity at $s=1$; this requires the asymptotics of the
incomplete gamma function. The manipulations involve polylogs for which we find
a representation in terms of Nielsen integrals, as well as mysterious
conjectures for Bernoulli numbers. Applications include the determination of
the asymptotic growth of the Taylor coefficients of $(-z/\log(1-z))^k$. We also
give the asymptotics of Stirling numbers of first kind and their formula in
terms of harmonic numbers.
