On the Hurwitz Zeta Function
By L. Fekih-Ahmed. 2011
97 citations recorded by the source corpus.
Abstract We give new integral and series representations of the Hurwitz zeta function.We also provide a closed-form expression of the coefficients of the Laurent expansionof the Hurwitz-zeta function about any point in the complex plane.Keywords Number Theory · Hurwitz Zeta functionMathematics Subject Classification (2000) 11M06 · 11M351 IntroductionThe Hurwitz zeta function defined by the seriesζ(s,a) =1a s +1(1+ a) s + ··· +1(k− 1 +a) s +··· =X ∞n=1 (n−1 + a) −s , (1.1)where 0 1, and can be analyticallycontinued to the whole complex plane with one singularity, a simple pole with residue1 at s= 1.The Hurwitz zeta function has also the following integral representationζ(s,a) =1Γ(s)Z ∞0 e −(a 1)t e t − 1t s−1 dt, (1.2)valid for ℜ(s) >1 [6]. Moreover, it has the following analytic continuation repre-sented by the following contour integralΓ(1 − s)2πiZ C e −(a−1)t e t − 1t s−1 dt, (1.3)where C is the Hankel contour consisting of the three parts C= C
Subfields: Algebra and Number Theory, Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/8f409d010c3b27c105a5223298d98e43e8bc979a
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In Algebra and Number Theory
In Applied Mathematics
Citations in the corpus
- Special functions related to Dedekind-type DC-sums and their applications (2009, background): “([2], [6], [16], [15], [14], [13], [24], [55], [56], [58], [29]) and see also the references cited in each of these earlier works.”
- A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations (2014, background): “(7) which was also derived by Berndt in In 1972, Berndt, being inspired by the similar Lammel’s proof for the Riemann ζ–function [43], also proved this formula in [9].”