The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function

Abstract

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_γ(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 γ_k= γ_k(1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions.