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Concept# Hurwitz zeta function

Summary

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
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.
The Hurwitz zeta function has an integral representation
for and (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing
and then interchanging the sum and integral.
The integral representation above can be converted to a contour integral representation
where is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation . Unlike the previous integral, this integral is valid for all s, and indeed is an entire function of s.
The contour integral representation provides an analytic continuation of to all . At , it has a simple pole with residue .
The Hurwitz zeta function satisfies an identity which generalizes the functional equation of the Riemann zeta function:
valid for Re(s) > 1 and 0 < a ≤ 1. The Riemann zeta functional equation is the special case a = 1:
Hurwitz's formula can also be expressed as
(for Re(s) < 0 and 0 < a ≤ 1).
Hurwitz's formula has a variety of different proofs. One proof uses the contour integration representation along with the residue theorem. A second proof uses a theta function identity, or equivalently Poisson summation. These proofs are analogous to the two proofs of the functional equation for the Riemann zeta function in Riemann's 1859 paper. Another proof of the Hurwitz formula uses Euler–Maclaurin summation to express the Hurwitz zeta function as an integral
(−1 < Re(s) < 0 and 0 < a ≤ 1) and then expanding the numerator as a Fourier series.
When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers ,
holds for all values of s.

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