In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hurwitz zeta function is one of the many
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
s. It is formally defined for
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
variables with and by
:
This series is
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
for the given values of and and can be extended to a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
defined for all . The
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is . The Hurwitz zeta function is named after
Adolf Hurwitz, who introduced it in 1882.
Integral representation
The Hurwitz zeta function has an integral representation
:
for
and
(This integral can be viewed as a
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
.) 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
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
Contour integration is closely related to the calculus of residues, a method of complex analysis.
...
representation
:
where
is a
Hankel contour
In mathematics, a Hankel contour is a path in the complex plane which extends from
(+∞,δ), around the origin counter clockwise and back to
(+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitraril ...
counterclockwise around the positive real axis, and the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are use ...
is used for the
complex exponentiation . Unlike the previous integral, this integral is valid for all ''s'', and indeed is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
of ''s''.
The contour integral representation provides an
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
of
to all
. At
, it has a
simple pole
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
with
residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
.
Hurwitz's formula
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
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as wel ...
.
A second proof uses a
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
identity, or equivalently
Poisson summation
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...
. 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
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
.
Functional equation for rational ''a''
When ''a'' is a rational number, Hurwitz's formula leads to the following
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
: For integers
,
:
holds for all values of ''s''.
This functional equation can be written as another equivalent form:
.
Some finite sums
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. Appendix B in.
Series representation
A convergent
Newton series
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
representation defined for (real) ''a'' > 0 and any complex ''s'' ≠ 1 was given by
Helmut Hasse in 1930:
:
This series converges uniformly on
compact subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
s of the ''s''-plane to an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
. The inner sum may be understood to be the ''n''th
forward difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
of
; that is,
:
where Δ is the
forward difference operator
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
. Thus, one may write:
:
Taylor series
The partial derivative of the zeta in the second argument is a
shift:
:
Thus, the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
can be written as:
:
Alternatively,
:
with
.
Closely related is the Stark–Keiper formula:
:
which holds for integer ''N'' and arbitrary ''s''. See also
Faulhaber's formula
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers
:\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p
as a (''p''&nb ...
for a similar relation on finite sums of powers of integers.
Laurent series
The
Laurent series
In mathematics, the Laurent series of a complex function f(z) 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 a Taylor series expansion c ...
expansion can be used to define generalized
Stieltjes constants that occur in the series
:
In particular, the constant term is given by
:
where
is the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and
is the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strict ...
. As a special case,
.
Discrete Fourier transform
The
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
of the Hurwitz zeta function with respect to the order ''s'' is the
Legendre chi function
In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by
\chi_\nu(z) = \sum_^\infty \frac.
As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is triviall ...
.
Particular values
Negative integers
The values of ''ζ''(''s'', ''a'') at ''s'' = 0, −1, −2, ... are related to the
Bernoulli polynomials:
:
For example, the
case gives
:
''s''-derivative
The
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
with respect to ''s'' at ''s'' = 0 is related to the Gamma function:
:
In particular,
The formula is due to
Lerch.
Relation to Jacobi theta function
If
is the Jacobi
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
, then
: