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 each qn 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>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 &gamma; 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 &pi;:


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

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| &lt; 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| &lt; 2. 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+\nu+1 \choose k} \left[\zeta(k+\nu+2)-1\right]

= \zeta(\nu+2)$$

where &nu; 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 = &minus;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 &ge; 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 = &zeta;(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 &ge; 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)\left(\zeta(n+2)-1\right)-1$$

Expressions in the form of p-series
Adamchik and Srivastava give
 * $$\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)$$

and


 * $$\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.

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