Rational zeta series
Encyclopedia
In mathematics
, a rational zeta series is the representation of an arbitrary real number
in terms of a series consisting of rational number
s and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by
where
is a rational number, the value m is held fixed, and 
is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.
For m=2, a number of interesting numbers have a simple expression as rational zeta series:
and
where γ is the Euler–Mascheroni constant
. The series
follows by summing the Gauss–Kuzmin distribution. There are also series for π:
and
being notable because of its fast convergence. This last series follows from the general identity
which in turn follows from the generating function
for the Bernoulli numbers
Adamchik and Srivastava give a similar series
for the polygamma function at z=1, which is
.
The above converges for |z|<1. A special case is
which holds for
. Here, ψ is the digamma function and 
is the polygamma function. Many series involving the binomial coefficient
may be derived:
where
is a complex number. The above follows from the series expansion for the Hurwitz zeta
taken at
. Similar series may be obtained by simple algebra:
and
and
For integer
, the series
can be written as the finite sum
The above follows from the simple recursion relation
. Next, the series
may be written as
for integer
. The above follows from the identity 
. This process may be applied recursively to obtain finite series for general expressions of the form
for positive integers m.
and
where
are the Bernoulli number
s and
are the Stirling numbers of the second kind.
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...
, a rational zeta series is the representation of an arbitrary real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
in terms of a series consisting of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by
where
is a rational number, the value m is held fixed, and is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.Elementary series
For integer m, one hasFor m=2, a number of interesting numbers have a simple expression as rational zeta series:
and
where γ is the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
. The series
follows by summing the Gauss–Kuzmin distribution. There are also series for π:
and
being notable because of its fast convergence. This last series follows from the general identity
which in turn follows from the generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
for the Bernoulli numbers
Adamchik and Srivastava give a similar series
Polygamma-related series
A number of additional relationships can be derived from the Taylor seriesTaylor 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....
for the polygamma function at z=1, which is
.The above converges for |z|<1. A special case is
which holds for
. Here, ψ is the digamma function and is the polygamma function. Many series involving the binomial coefficientBinomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
may be derived:
where
is a complex number. The above follows from the series expansion for the Hurwitz zetataken at
. Similar series may be obtained by simple algebra:and
and
For integer
, the seriescan be written as the finite sum
The above follows from the simple recursion relation
. Next, the seriesmay be written as
for integer
. The above follows from the identity . This process may be applied recursively to obtain finite series for general expressions of the formfor positive integers m.
Half-integer power series
Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one hasExpressions in the form of p-series
Adamchik and Srivastava giveand
where
are the Bernoulli numberBernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s and
are the Stirling numbers of the second kind.Other series
Other constants that have notable rational zeta series are:- Khinchin's constant
- Apéry's constantApéry's constantIn mathematics, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics...