Hurwitz zeta function

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In mathematics, the Hurwitz zeta function is one of the many zeta functions. It defined as

<math>\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s}.</math>

When q = 1, this coincides with Riemann's zeta function. The Dirichlet L-functions may be expressed as a linear combination of the Hurwitz zeta function, and thus the study of L-functions can be unified to a study of the Hurwitz zeta function.

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Analytic structure

The Hurwitz zeta function has a simple pole at s=1, with residue 1. The constant term is given by

<math>\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] =

\frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)</math>

where <math>\Gamma</math> is the Gamma function and ψ is the digamma function.

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Series representation

The Hurwitz zeta function has a globally convergent series representation defined on the entire complex s-plane, excluding s=1. Due to Helmut Hasse in 1930, it is

<math>\zeta(s,q)=\frac{1}{s-1}

\sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.</math>

This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the n 'th forward difference of <math>q^{1-s}</math>; that is,

<math>\Delta^n q^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (q+k)^{1-s}</math>

where Δ is the forward difference operator. Thus, one may write

<math>\zeta(s,q)=\frac{1}{s-1}

\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}</math>.

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Hurwitz's formula

Hurwitz's formula is the theorem that

<math>\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]</math>

where

<math>\beta(x;s)=

2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}= \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix}) </math> is a representation of the zeta that is valid for <math>0\le x\le 1</math> and <math>s>1</math>. Here, <math>\mbox{Li}_s (z)</math> is the polylogarithm.

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Relation to Bernoulli polynomials

The function <math>\beta</math> defined above generalizes the Bernoulli polynomials:

<math>B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right] </math>

where <math>\Re z</math> denotes the real part of z. Alternately,

<math>\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.</math>
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Relation to the polygamma function

The Hurwitz zeta generalizes the polygamma function:

<math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z).\,</math>
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Relation to the Lerch transcendent

The Lerch transcendent generalizes the Hurwitz zeta:

<math>\Phi(z, s, q) = \sum_{k=0}^\infty

\frac { z^k} {(k+q)^s}</math> and thus

<math>\zeta (s,q)=\Phi(1, s, q).\,</math>
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Functional equation

The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers <math>1\leq m \leq n </math>,

<math>\zeta \left(1-s,\frac{m}{n} \right) =

\frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\; \zeta \left( s,\frac {k}{n} \right) </math> holds for all values of s.

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Taylor series

The derivative of the zeta in the second argument is a shift:

<math>\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).</math>

Thus, the Taylor series can be written as

<math>\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!}

\frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).</math>

Closely related is the Stark-Keiper formula:

<math>\zeta(s,N) =

\sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right] {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N) </math>

which holds for integer N and arbitrary s.

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Fourier transform

The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function.

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Multiplication theorem

The multiplication theorem is given by

<math>\sum_{k=1}^n \zeta\left(s, \frac{k}{n}\right) = n^s\zeta(s)</math>
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Relation to Jacobi theta function

If <math>\vartheta (z,\tau)</math> is the Jacobi theta function, then

<math>\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}=

\pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]</math>

holds for <math>\Re s > 0</math> and z complex, but not an integer. For z=n an integer, this simplifies to

<math>\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}=

2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).</math>

where ζ here is the Riemann zeta function. Note that this later form is the functional equation for the Riemann zeta, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the Dirac delta function in z as <math>t\rightarrow 0</math>.

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Rational values

The zeta function is engaged in a number of striking identities at rational values (given by Djurdje Cvijović and Jacek Klinowski, reference below). In particular, values in terms of the Euler polynomials <math>E_n(x)</math>:

<math>E_{2n-1}\left(\frac{p}{q}\right) =

(-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}} \sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right) \cos \frac{(2k-1)\pi p}{q}</math>

and

<math>E_{2n}\left(\frac{p}{q}\right) =

(-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}} \sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right) \sin \frac{(2k-1)\pi p}{q}</math>

One also has

<math>\zeta\left(s,\frac{2p-1}{2q}\right) =

2(2q)^{s-1} \sum_{k=1}^q \left[ C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) + S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right) \right]</math>

which holds for <math>1\le p \le q</math>. Here, the <math>C_\nu(x)</math> and <math>S_\nu(x)</math> are defined by means of the Legendre chi function <math>\chi_\nu</math> as

<math>C_\nu(x) = \mbox{Re} \chi_\nu (e^{ix})</math>

and

<math>S_\nu(x) = \mbox{Im} \chi_\nu (e^{ix})</math>

For integer values of ν, these may be expressed in terms of the Euler polynomials (which see). These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

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Applications

Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics, it occurs in Zipf's law and the Zipf-Mandelbrot law. In particle physics, it occurs in a formula by Julian Schwinger, given in 1951, giving an exact result for the pair production rate in a uniform electric field.

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References

it:Funzione zeta di Hurwitz

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