  The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the
Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and
are of fundamental and long-standing importance in analytic number theory and
other disciplines. We present an array of exact results for the Stieltjes
coefficients, including series representations and summatory relations. Other
integral representations provide the difference of Stieltjes coefficients at
rational arguments. The presentation serves to link a variety of topics in
analysis and special function and special number theory, including logarithmic
series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet
$L$-functions at special points. The results have a wide range of application,
both theoretical and computational.
