The Stieltjes constants γk(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 γk(1) to the ηj 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 S2(n) = Sγ(n) + SΛ(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum Sγ(n) is O(n) and we present various integral representations for it. We present novel series representations of S2(n). We additionally present a rapidly convergent expression for γk = γk(1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions.