In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Adolf Hurwitz ( 26 March 1859 - 18 November 1919) (ˈadɒlf ˈhurvits was a German mathematician and was described by Jean-Pierre A zeta function is a function which is composed of an infinite sum of powers that is which may be written as a Dirichlet series: \zeta(s = \sum_{k=1}^{\infty}f(k^s It is formally defined for complex arguments s with Re(s)>1 and q with Re(q)>0 by

$\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s}.$

This series is absolutely convergent for the given values of s and q and can be extended to a meromorphic function defined for all s≠1. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic The Riemann zeta function is ζ(s,1). In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in

## Analytic continuation

The Hurwitz zeta function can be extended by analytic continuation to a meromorphic function defined for all complex numbers s with s ≠ 1. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic At s = 1 it has a simple pole with residue 1. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Complex analysis, the residue is a Complex number which describes the behavior of Line integrals of a Meromorphic function around a singularity The constant term is given by

$\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] = \frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)$

where Γ is the Gamma function and ψ is the digamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function In Mathematics, the digamma function is defined as the Logarithmic derivative of the Gamma function: \psi(x =\frac{d}{dx} \ln{\Gamma(x}=

## Series representation

A convergent series representation defined for q > −1 and any complex s ≠ 1 was given by Helmut Hasse in 1930 [1]:

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1}\sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.$

This series converges uniformly on compact subsets of the s-plane to an entire function. Helmut Hasse (ˈhasə ( 25 August 1898 – 26 December 1979) was a German Mathematician working in Algebraic In Complex analysis, an entire function, also called an integral function is a complex-valued function that is holomorphic everywhere on the The inner sum may be understood to be the nth forward difference of q1 − s; that is,

$\Delta^n q^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (q+k)^{1-s}$

where Δ is the forward difference operator. A finite difference is a mathematical expression of the form f ( x + b) &minus f ( x + a) In Mathematics, a difference operator maps a function, f ( x) to another function f ( x + a) &minus f ( x Thus, one may write

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}$
$= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} q^{1-s}.$

## Integral representation

The function has an integral representation in terms of the Mellin transform as

$\zeta(s,q)=\frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt$

for $\Re s>1$ and $\Re q >0.$

## Hurwitz's formula

Hurwitz's formula is the theorem that

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

where

$\beta(x;s)=2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}=\frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix})$

is a representation of the zeta that is valid for $0\le x\le 1$ and s > 1. In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform Here, Lis(z) is the polylogarithm. The polylogarithm (also known as de Jonquière's function) is a Special function Li s ( z) that is defined by the sum

## Functional equation

The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values For integers $1\leq m \leq n$,

$\zeta \left(1-s,\frac{m}{n} \right) = \frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\;\zeta \left( s,\frac {k}{n} \right)$

holds for all values of s.

## Taylor series

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

$\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).$

Thus, the Taylor series has the distinctly umbral form:

$\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!} \frac {\partial^k} {\partial x^k} \zeta (s,x) =\sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).$

Closely related is the Stark-Keiper formula:

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

which holds for integer N and arbitrary s. In Mathematics, a Sheffer sequence is a Polynomial sequence, i In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics before the 1970s the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations and See also Faulhaber's formula for a similar relation on finite sums of powers of integers. In Mathematics, Faulhaber's formula, named after Johann Faulhaber, expresses the sum \sum_{k=1}^n k^p = 1^p + 2^p + 3^p + \cdots + n^p

## Fourier transform

The discrete Fourier transform of the Hurwitz zeta function with respect to the order s is the Legendre chi function. In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis. In Mathematics, the Legendre chi function is a Special function whose Taylor series is also a Dirichlet series, given by

## Relation to Bernoulli polynomials

The function β defined above generalizes the Bernoulli polynomials:

$B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right]$

where $\Re z$ denotes the real part of z. In Mathematics, the Bernoulli polynomials occur in the study of many Special functions and in particular the Riemann zeta function and the Hurwitz Alternately,

$\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.$

In particular, the relation holds for n = 0 and one has

$\zeta(0,x)= \frac{1}{2} -x$

## Relation to Jacobi theta function

If $\vartheta (z,\tau)$ is the Jacobi theta function, then

$\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}= \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]$

holds for $\Re s > 0$ and z complex, but not an integer. In Mathematics, theta functions are Special functions of Several complex variables. For z=n an integer, this simplifies to

$\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}= 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s)=2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).$

where ζ here is the Riemann zeta function. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values 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 $t\rightarrow 0$. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac.

## Relation to Dirichlet L-functions

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 q=1, when q=1/2 it is equal to (2s-1)ζ(s), and if q=n/k with k>2, (n,k)>1 and 0<n<k, then

$\zeta(s,n/k)=\sum_\chi\overline{\chi}(n)L(s,\chi),$

the sum running over all Dirichlet characters mod k. In mathematics a Dirichlet L -series, named in honour of Johann Peter Gustav Lejeune Dirichlet, is a function of the form L(s\chi = \sum_{n=1}^\infty In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In Number theory, Dirichlet characters are certain Arithmetic functions which arise from Completely multiplicative characters on the units of In the opposite direction we have the linear combination

$L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).$

There is also the multiplication theorem

$k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),$

of which a useful generalization is

$\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).$

(This last form is valid whenever q a natural number and 1-qa is not. In Mathematics, the multiplication theorem is a certain type of identity obeyed by many Special functions related to the Gamma function. )

## Zeros

If q=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if q=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<q<1 and q≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn[2] for rational and non-algebraic irrational q and by Cassels[3] for algebraic irrational q. Harold Davenport ( 30 October 1907 – 9 June 1969) was an English mathematician known for his extensive work in Number theory Hans Arnold Heilbronn ( 8 October 1908, Berlin – 28 April 1975, Toronto) was a mathematician John William Scott Cassels FRS (born July 11, 1922 in Durham) is a leading English Mathematician.

## Rational values

The Hurwitz zeta function occurs 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 En(x):

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

and

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

One also has

$\zeta\left(s,\frac{2p-1}{2q}\right) = 2(2q)^{s-1} \sum_{k=1}^q \left[C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) +S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right) \right]$

which holds for $1\le p \le q$. In Mathematics, the Bernoulli polynomials occur in the study of many Special functions and in particular the Riemann zeta function and the Hurwitz Here, the Cν(x) and Sν(x) are defined by means of the Legendre chi function χν as

$C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})$

and

$S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).$

For integer values of ν, these may be expressed in terms of the Euler polynomials. In Mathematics, the Legendre chi function is a Special function whose Taylor series is also a Dirichlet series, given by In Mathematics, the Bernoulli polynomials occur in the study of many Special functions and in particular the Riemann zeta function and the Hurwitz These relations may be derived by employing the functional equation together with Hurwitz's formula, given above.

## 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. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes However, it also occurs in the study of fractals and dynamical systems. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential In applied statistics, it occurs in Zipf's law and the Zipf-Mandelbrot law. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one In particle physics, it occurs in a formula by Julian Schwinger[4], giving an exact result for the pair production rate of a Dirac electron in a uniform electric field. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Julian Seymour Schwinger ( February 12, 1918 &ndash July 16, 1994) was an American Theoretical physicist. See also Electron-positron annihilation Meitner–Hupfeld effect Pair instability supernova In Physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides

## Special cases and generalizations

The Hurwitz zeta function generalizes the polygamma function:

ψ(m)(z) = ( − 1)m + 1m!ζ(m + 1,z). In Mathematics, the polygamma function of order m is defined as the ( m  + 1th Derivative of the logarithm of the Gamma function

The Lerch transcendent generalizes the Hurwitz zeta:

$\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}$

and thus

$\zeta (s,q)=\Phi(1, s, q).\,$

## References

1. ^ Helmut Hasse, Ein Summierungsverfahren fur die Riemannsche ζ-Reihe, (1930) Math. In Mathematics, the Lerch zeta-function, sometimes called the Hurwitz-Lerch zeta-function, is a Special function that generalizes the Hurwitz Z. 32 pp 458-464.
2. ^ Davenport, H. and Heilbronn, H. On the zeros of certain Dirichlet series J. London Math. Soc. 11 (1936), pp. 181-185
3. ^ Cassels, J. W. S. Footnote to a note of Davenport and Heilbronn J. London Math. Soc. 36 (1961), pp. 177-184
4. ^ Schwinger, J. , On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951), pp. 664-679.

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