On representations and differences of Stieltjes coefficients, and other relations

Details On representations and differences of Stieltjes coefficients, and other relations On representations and differences of Stieltjes coefficients, and other relations

2008-09-19

arxiv.org/abs/0809.3277v2

Physics

Mathematical Physics

39 pages, no figures Prop. 8 strengthened

Scientific paper

The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and computational.

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