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

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

2007-06-03

arxiv.org/abs/0706.0343v2

Physics

Mathematical Physics

37 pages, no figures Prop. 3(b) added and minor updates

Scientific paper

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.

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