Hurwitz zeta function

[1] The Hurwitz zeta function has an integral representation for

The formula can be obtained, roughly, by writing and then interchanging the sum and integral.

is a Hankel contour counterclockwise around the positive real axis, and the principal branch is used for the complex exponentiation

The Riemann zeta functional equation is the special case a = 1:[7] Hurwitz's formula can also be expressed as[8] (for Re(s) < 0 and 0 < a ≤ 1).

Hurwitz's formula has a variety of different proofs.

[9] One proof uses the contour integration representation along with the residue theorem.

[6][8] A second proof uses a theta function identity, or equivalently Poisson summation.

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.

[11] When a is a rational number, Hurwitz's formula leads to the following functional equation: For integers

, holds for all values of s.[12] This functional equation can be written as another equivalent form:

Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form where m is positive integer greater than 2 and s is complex, see e.g.

[13] A convergent Newton series representation defined for (real) a > 0 and any complex s ≠ 1 was given by Helmut Hasse in 1930:[14] This series converges uniformly on compact subsets of the s-plane to an entire function.

The inner sum may be understood to be the nth forward difference of

Thus, one may write: The partial derivative of the zeta in the second argument is a shift: Thus, the Taylor series can be written as: Alternatively, with

[15] Closely related is the Stark–Keiper formula: which holds for integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers.

[16] The values of ζ(s, a) at s = 0, −1, −2, ... are related to the Bernoulli polynomials:[17] For example, the

case gives[18] The partial derivative with respect to s at s = 0 is related to the gamma function: In particular,

For z=n an integer, this simplifies to where ζ here is the Riemann zeta function.

The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic delta function, or Dirac comb in z as

At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when a = 1, when a = 1/2 it is equal to (2s−1)ζ(s),[21] and if a = n/k with k > 2, (n,k) > 1 and 0 < n < k, then[22] the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination[21] There is also the multiplication theorem of which a useful generalization is the distribution relation[23] (This last form is valid whenever q a natural number and 1 − qa is not.)

In particular, there will be no zeros with real part greater than or equal to 1.

However, if 0

[21][25] The Hurwitz zeta function occurs in a number of striking identities at rational values.

are defined by means of the Legendre chi function

as and For integer values of ν, these may be expressed in terms of the Euler polynomials.

These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

Hurwitz's zeta function occurs in a variety of disciplines.

However, it also occurs in the study of fractals and dynamical systems.

In particle physics, it occurs in a formula by Julian Schwinger,[27] giving an exact result for the pair production rate of a Dirac electron in a uniform electric field.

The Lerch transcendent generalizes the Hurwitz zeta: and thus Hypergeometric function Meijer G-function