Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by

ζ ( s , a ) = ∑ n = 0 ∞ 1 ( n + a ) s . {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.}

This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s, 1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.

Integral representation The Hurwitz zeta function has an integral representation

ζ ( s , a ) = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e − a x 1 − e − x d x {\displaystyle \zeta (s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-ax}}{1-e^{-x}}}dx}

for Re ⁡ ( s ) > 1 {\displaystyle \operatorname {Re} (s)>1} and Re ⁡ ( a ) > 0. {\displaystyle \operatorname {Re} (a)>0.} (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing

ζ ( s , a ) Γ ( s ) = ∑ n = 0 ∞ 1 ( n + a ) s ∫ 0 ∞ x s e − x d x x = ∑ n = 0 ∞ ∫ 0 ∞ y s e − ( n + a ) y d y y {\displaystyle {\begin{aligned}\zeta (s,a)\Gamma (s)&=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}\\[1ex]&=\sum _{n=0}^{\infty }\int _{0}^{\infty }y^{s}e^{-(n+a)y}{\frac {dy}{y}}\end{aligned}}}

and then interchanging the sum and integral. The integral representation above can be converted to a contour integral representation

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