  The Stieltjes constants gamma_k(a) are the expansion coefficients in the
Laurent series for the Hurwitz zeta function about its only pole at s=1. We
present the relation of gamma_k(1) to the eta_j coefficients that appear in the
Laurent expansion of the logarithmic derivative of the Riemann zeta function
about its pole at s=1. We obtain novel integral representations of the
Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) +
S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the
Riemann hypothesis. The sum S_\gamma(n) is O(n) and we present various integral
representations for it. We present novel series representations of S_2(n). We
additionally present a rapidly convergent expression for \gamma_k= \gamma_k(1)
and a variety of results pertinent to a parameterized representation of the
Riemann and Hurwitz zeta functions.
