# 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 qn is a rational number, the value m is held fixed, and ζ(sm) 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>1, one has

$x=\sum _{n=2}^{\infty }q_{n}\left[\zeta (n)-\sum _{k=1}^{m-1}k^{-n}\right]$

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

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

and

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

where γ is the Euler–Mascheroni constant. The series

$\log 2=\sum _{n=1}^{\infty }{\frac {1}{n}}\left[\zeta (2n)-1\right]$

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

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

and

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

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

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

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

${\frac {t}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {t^{n}}{n!}}$

Adamchik and Srivastava give a similar series

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

## 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

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

which holds for |t| < 2. Here, ψ is the digamma function and ψ(m) is the polygamma function. Many series involving the binomial coefficient may be derived:

$\sum _{k=0}^{\infty }{k+\nu +1 \choose k}\left[\zeta (k+\nu +2)-1\right]=\zeta (\nu +2)$

where ν 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:

$\sum _{k=0}^{\infty }{k+\nu +1 \choose k+1}\left[\zeta (k+\nu +2)-1\right]=1$

and

$\sum _{k=0}^{\infty }(-1)^{k}{k+\nu +1 \choose k+1}\left[\zeta (k+\nu +2)-1\right]=2^{-(\nu +1)}$

and

$\sum _{k=0}^{\infty }(-1)^{k}{k+\nu +1 \choose k+2}\left[\zeta (k+\nu +2)-1\right]=\nu \left[\zeta (\nu +1)-1\right]-2^{-\nu }$

and

$\sum _{k=0}^{\infty }(-1)^{k}{k+\nu +1 \choose k}\left[\zeta (k+\nu +2)-1\right]=\zeta (\nu +2)-1-2^{-(\nu +2)}$

For integer n ≥ 0, the series

$S_{n}=\sum _{k=0}^{\infty }{k+n \choose k}\left[\zeta (k+n+2)-1\right]$

can be written as the finite sum

$S_{n}=(-1)^{n}\left[1+\sum _{k=1}^{n}\zeta (k+1)\right]$

The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series

$T_{n}=\sum _{k=0}^{\infty }{k+n-1 \choose k}\left[\zeta (k+n+2)-1\right]$

may be written as

$T_{n}=(-1)^{n+1}\left[n+1-\zeta (2)+\sum _{k=1}^{n-1}(-1)^{k}(n-k)\zeta (k+1)\right]$

for integer n ≥ 1. The above follows from the identity Tn + Tn + 1 = Sn. This process may be applied recursively to obtain finite series for general expressions of the form

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

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

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

## Expressions in the form of p-series

$\sum _{n=2}^{\infty }n^{m}\left[\zeta (n)-1\right]=1\,+\sum _{k=1}^{m}k!\;S(m+1,k+1)\zeta (k+1)$
$\sum _{n=2}^{\infty }(-1)^{n}n^{m}\left[\zeta (n)-1\right]=-1\,+\,{\frac {1-2^{m+1}}{m+1}}B_{m+1}\,-\sum _{k=1}^{m}(-1)^{k}k!\;S(m+1,k+1)\zeta (k+1)$
where $B_{k}$  are the Bernoulli numbers and $S(m,k)$  are the Stirling numbers of the second kind.