  The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the
Laurent expansion for the Hurwitz zeta function $\zeta(s,a)$. We present
summatory results for these constants $\gamma_k(a)$ in terms of fundamental
mathematical constants such as the Catalan constant, and further relate them to
products of rational functions of prime numbers. We provide examples of
infinite series of differences of Stieltjes constants evaluating as volumes in
hyperbolic $3$-space. We present a new series representation for the difference
of the first Stieltjes constant at rational arguments. We obtain expressions
for $\zeta(1/2)L_{-p}(1/2)$, where for primes $p>7$, $L_{-p}(s)$ are certain
$L$-series, and remarkably tight bounds for the value $\zeta(1/2)$,
$\zeta(s)=\zeta(s,1)$ being the Riemann zeta function.
