Hurwitz zeta function
Encyclopedia
In mathematics
, the Hurwitz zeta function, named after Adolf Hurwitz
, is one of the many zeta functions. It is formally defined for complex
arguments s with Re(s) > 1 and q with Re(q) > 0 by
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. The Riemann zeta function is ζ(s,1).
where the contour
C is a loop around the negative real axis. This provides an analytic continuation of
.
The Hurwitz zeta function can be extended by analytic continuation
to a meromorphic function
defined for all complex numbers s with s ≠ 1. At s = 1 it has a simple pole with residue
1. The constant term is given by
where Γ is the Gamma function
and ψ is the digamma function.
in 1930:
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
; that is,
where Δ is the forward difference operator. Thus, one may write
as
for
and 
where
is a representation of the zeta that is valid for
and s > 1. Here, 
is the polylogarithm.
relates values of the zeta on the left- and right-hand sides of the complex plane. For integers
,
holds for all values of s.
Thus, the Taylor series
has the distinctly umbral
form:
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.
expansion can be used to define Stieltjes constants that occur in the series
of the Hurwitz zeta function with respect to the order s is the Legendre chi function.
defined above generalizes the Bernoulli polynomials
:
where
denotes the real part of z. Alternately,
In particular, the relation holds for
and one has
is the Jacobi theta function, then
holds for
and z complex, but not an integer. For z=n an integer, this simplifies to
where ζ here is the Riemann zeta function. Note that this latter form is the functional equation
for the Riemann zeta function, 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
.
the sum running over all Dirichlet character
s mod k. In the opposite direction we have the linear combination
There is also the multiplication theorem
of which a useful generalization is
(This last form is valid whenever q a natural number and 1 − qa is not.)
s)<1+ε for any positive real number ε. This was proved by Davenport
and Heilbronn
for rational and non-algebraic irrational q, and by Cassels
for algebraic irrational q.
:
and
One also has
which holds for
. Here, the 
and 
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.
, 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
, giving an exact result for the pair production
rate of a Dirac
electron in a uniform electric field.
The Barnes zeta function
generalizes the Hurwitz zeta function.
The Lerch transcendent generalizes the Hurwitz zeta:
and thus
Hypergeometric function
where 
Meijer G-function
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hurwitz zeta function, named after Adolf Hurwitz
Adolf Hurwitz
Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...
, is one of the many zeta functions. It is formally defined for complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
arguments s with Re(s) > 1 and q with Re(q) > 0 by
This series is absolutely convergent for the given values of s and q and can be extended to a meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
defined for all s≠1. The Riemann zeta function is ζ(s,1).
Analytic continuation
If Re(s) ≤ 1 the Hurwitz zeta function can be defined by the equationwhere the contour
Contour
Contour may refer to:* an outline or silhouette* a contour line on a contour map, or the corresponding line on the ground or sea bed* Contour , a phonetic sound* Pitch contour , a melody shape...
C is a loop around the negative real axis. This provides an analytic continuation of
.The Hurwitz zeta function can be extended by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
to a meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
defined for all complex numbers s with s ≠ 1. At s = 1 it has a simple pole with residue
Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
1. The constant term is given by
where Γ is the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and ψ is the digamma function.
Series representation
A convergent series representation defined for q > −1 and any complex s ≠ 1 was given by Helmut HasseHelmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
in 1930:
This series converges uniformly on compact subsets of the s-plane to an entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
. The inner sum may be understood to be the nth forward difference of
; that is,where Δ is the forward difference operator. Thus, one may write
Integral representation
The function has an integral representation in terms of the Mellin transformMellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
as
for
and Hurwitz's formula
Hurwitz's formula is the theorem thatwhere
is a representation of the zeta that is valid for
and s > 1. Here, is the polylogarithm.Functional equation
The functional equationFunctional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
relates values of the zeta on the left- and right-hand sides of the complex plane. For integers
,holds for all values of s.
Taylor series
The derivative of the zeta in the second argument is a shift:Thus, the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
has the distinctly umbral
Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method...
form:
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.
Laurent series
The Laurent seriesLaurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
expansion can be used to define Stieltjes constants that occur in the series
Fourier transform
The discrete Fourier transformDiscrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
of the Hurwitz zeta function with respect to the order s is the Legendre chi function.
Relation to Bernoulli polynomials
The function defined above generalizes the Bernoulli polynomialsBernoulli polynomials
In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator...
:
where
denotes the real part of z. Alternately,In particular, the relation holds for
and one hasRelation to Jacobi theta function
If is the Jacobi theta function, thenholds for
and z complex, but not an integer. For z=n an integer, this simplifies towhere ζ here is the Riemann zeta function. Note that this latter form is the functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
for the Riemann zeta function, 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
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
in z as
.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, thenthe sum running over all Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
s mod k. In the opposite direction we have the linear combination
There is also the multiplication theorem
Multiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name...
of which a useful generalization is
(This last form is valid whenever q a natural number and 1 − qa is not.)
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 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 1Harold Davenport
Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...
and Heilbronn
Hans Heilbronn
Hans Arnold Heilbronn was a mathematician.He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate...
for rational and non-algebraic irrational q, and by Cassels
J. W. S. Cassels
John William Scott Cassels , FRS is a leading English mathematician.-Biography:Educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh, Cassels graduated from the University of Edinburgh with an MA in 1943.His academic career was interrupted in World War II...
for algebraic irrational q.
Rational values
The Hurwitz zeta function occurs in a number of striking identities at rational values. In particular, values in terms of the Euler polynomials:and
One also has
which holds for
. Here, the and are defined by means of the Legendre chi function asand
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.
Applications
Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theoryNumber theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, it occurs in Zipf's law and the Zipf–Mandelbrot law. In particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, it occurs in a formula by Julian Schwinger
Julian Schwinger
Julian Seymour Schwinger was an American theoretical physicist. He is best known for his work on the theory of quantum electrodynamics, in particular for developing a relativistically invariant perturbation theory, and for renormalizing QED to one loop order.Schwinger is recognized as one of the...
, giving an exact result for the pair production
Pair production
Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon . For example an electron and its antiparticle, the positron, may be created...
rate of a Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
electron in a uniform electric field.
Special cases and generalizations
The Hurwitz zeta function is related to the polygamma function:The Barnes zeta function
Barnes zeta function
In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function.-Definition:The Barnes zeta function is defined by...
generalizes the Hurwitz zeta function.
The Lerch transcendent generalizes the Hurwitz zeta:
and thus
Hypergeometric function
where Meijer G-function
Meijer G-Function
In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...