Rational zeta series

In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers 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

:$x=sum\_\{n=2\}^infty\; q\_n\; zeta\; (n,m)$

where $q\_n$ is a rational number, the value "m" is held fixed, and $zeta(s,m)$ 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 has

:$x=sum\_\{n=2\}^infty\; q\_n\; left\; [zeta(n)-\; sum\_\{k=1\}^\{m-1\}\; k^\{-n\}\; ight]$

For "m=2", a number of interesting numbers have a simple expression as rational zeta series:

:$1=sum\_\{n=2\}^infty\; left\; [zeta(n)-1\; ight]$

and:$1-gamma=sum\_\{n=2\}^infty\; frac\{1\}\{n\}left\; [zeta(n)-1\; ight]$

where γ is the Euler-Mascheroni constant. The series:$log\; 2\; =sum\_\{n=1\}^infty\; frac\{1\}\{n\}left\; [zeta(2n)-1\; ight]$

follows by summing the Gauss-Kuzmin distribution. There are also series for π:

:$log\; pi\; =sum\_\{n=2\}^infty\; frac\{2(3/2)^n-3\}\{n\}left\; [zeta(n)-1\; ight]$

and

:$frac\{13\}\{30\}\; -\; frac\{pi\}\{8\}\; =sum\_\{n=1\}^infty\; frac\{1\}\{4^\{2nleft\; [zeta(2n)-1\; ight]$

being notable because of its fast convergence. This last series follows from the general identity

:$sum\_\{n=1\}^infty\; (-1)^\{n\}\; t^\{2n\}\; left\; [zeta(2n)-1\; ight]\; =frac\{t^2\}\{1+t^2\}\; +\; frac\{1-pi\; t\}\{2\}\; -\; frac\; \{pi\; t\}\{e^\{2pi\; t\}\; -1\}$

which in turn follows from the generating function for the Bernoulli numbers

:$frac\{x\}\{e^x-1\}\; =\; sum\_\{n=0\}^infty\; B\_n\; frac\{t^n\}\{n!\}$

Adamchik and Srivastava give a similar series

:$sum\_\{n=1\}^infty\; frac\{t^\{2n\{n\}\; zeta(2n)\; =\; log\; left(frac\{pi\; t\}\; \{sin\; (pi\; t)\}\; ight)$

Polygamma-related series

A number of additional relationships can be derived from the Taylor series for the polygamma function at "z"=1, which is :$psi^\{(m)\}(z+1)=\; sum\_\{k=0\}^infty\; (-1)^\{m+k+1\}\; (m+k)!;\; zeta\; (m+k+1);\; frac\; \{z^k\}\{k!\}$.The above converges for |"z"|<1. A special case is

:$sum\_\{n=2\}^infty\; t^n\; left\; [zeta(n)-1\; ight]\; =\; -tleft\; [gamma\; +psi(1-t)\; -frac\{t\}\{1-t\}\; ight]$

which holds for . Here, &psi; is the digamma function and $psi^\{(m)\}$ is the polygamma function. Many series involving the binomial coefficient may be derived:

:$sum\_\{k=0\}^infty\; \{k+\; u+1\; choose\; k\}\; left\; [zeta(k+\; u+2)-1\; ight]\; =\; zeta(\; u+2)$

where $u$ is a complex number. The above follows from the series expansion for the Hurwitz zeta

:$zeta(s,x+y)\; =\; sum\_\{k=0\}^infty\; \{s+k-1\; choose\; s-1\}\; (-y)^k\; zeta\; (s+k,x)$taken at $y=-1$. Similar series may be obtained by simple algebra:

:$sum\_\{k=0\}^infty\; \{k+\; u+1\; choose\; k+1\}\; left\; [zeta(k+\; u+2)-1\; ight]\; =\; 1$

and

:$sum\_\{k=0\}^infty\; (-1)^k\; \{k+\; u+1\; choose\; k+1\}\; left\; [zeta(k+\; u+2)-1\; ight]\; =\; 2^\{-(\; u+1)\}$

and

:$sum\_\{k=0\}^infty\; (-1)^k\; \{k+\; u+1\; choose\; k+2\}\; left\; [zeta(k+\; u+2)-1\; ight]\; =\; u\; left\; [zeta(\; u+1)-1\; ight]\; -\; 2^\{-\; u\}$ and

:$sum\_\{k=0\}^infty\; (-1)^k\; \{k+\; u+1\; choose\; k\}\; left\; [zeta(k+\; u+2)-1\; ight]\; =\; zeta(\; u+2)-1\; -\; 2^\{-(\; u+2)\}$

For integer $ngeq\; 0$, the series:$S\_n\; =\; sum\_\{k=0\}^infty\; \{k+n\; choose\; k\}\; left\; [zeta(k+n+2)-1\; ight]$

can be written as the finite sum

:$S\_n=(-1)^nleft\; [1+sum\_\{k=1\}^n\; zeta(k+1)\; ight]$

The above follows from the simple recursion relation $S\_n+S\_\{n+1\}\; =\; zeta(n+2)$. Next, the series

:$T\_n\; =\; sum\_\{k=0\}^infty\; \{k+n-1\; choose\; k\}\; left\; [zeta(k+n+2)-1\; ight]$

may be written as

:$T\_n=(-1)^\{n+1\}left\; [n+1-zeta(2)+sum\_\{k=1\}^\{n-1\}\; (-1)^k\; (n-k)\; zeta(k+1)\; ight]$

for integer $ngeq\; 1$. The above follows from the identity $T\_n+T\_\{n+1\}\; =\; S\_n$. This process may be applied recursively to obtain finite series for general expressions of the form

:$sum\_\{k=0\}^infty\; \{k+n-m\; choose\; k\}\; left\; [zeta(k+n+2)-1\; ight]$

for 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 has

:$sum\_\{k=0\}^infty\; frac\; \{zeta(k+n+2)-1\}\{2^k\}$n+k+1} choose {n+1=left(2^{n+2}-1 ight)zeta(n+2)-1

Expressions in the form of p-series

Adamchik and Srivastava give :$sum\_\{n=2\}^infty\; n^m\; left\; [zeta(n)-1\; ight]\; =1,\; +\; sum\_\{k=1\}^m\; k!;\; S(m+1,k+1)\; zeta(k+1)$

and

:$sum\_\{n=2\}^infty\; (-1)^n\; n^m\; left\; [zeta(n)-1\; ight]\; =-1,\; +,\; frac\; \{1-2^\{m+1\{m+1\}\; B\_\{m+1\}\; ,-\; sum\_\{k=1\}^m\; (-1)^k\; k!;\; S(m+1,k+1)\; zeta(k+1)$

where $B\_k$ are the Bernoulli numbers and $S(m,k)$ are the Stirling numbers of the second kind.

Other series

Other constants that have notable rational zeta series are:
* Khinchin's constant
* Apéry's constant

References

* cite journal|author=Jonathan M. Borwein, David M. Bradley, Richard E. Crandall
title= [http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf Computational Strategies for the Riemann Zeta Function]
journal=J. Comp. App. Math.
year=2000
volume=121
pages=p.11
* cite journal|author=Victor S. Adamchik and H. M. Srivastava
title= [http://www-2.cs.cmu.edu/~adamchik/articles/sums/zeta.pdf Some series of the zeta and related functions]
journal=Analysis
year=1998
volume=18
pages=pp. 131–144