On the Hurwitz Zeta Function

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