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.

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Elementary series

For integer m, 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%2Bt^2} %2B \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{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)}\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%2B1)= \sum_{k=0}^\infty (-1)^{m%2Bk%2B1} (m%2Bk)!\; \zeta (m%2Bk%2B1)\; \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 %2B\psi(1-t) -\frac{t}{1-t}\right]

which holds for |t|<2. Here, ψ 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%2B\nu%2B1 \choose k} \left[\zeta(k%2B\nu%2B2)-1\right] = \zeta(\nu%2B2)

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

\zeta(s,x%2By) = \sum_{k=0}^\infty {s%2Bk-1 \choose s-1} (-y)^k \zeta (s%2Bk,x)

taken at y=-1. Similar series may be obtained by simple algebra:

\sum_{k=0}^\infty {k%2B\nu%2B1 \choose k%2B1} \left[\zeta(k%2B\nu%2B2)-1\right] = 1

and

\sum_{k=0}^\infty (-1)^k {k%2B\nu%2B1 \choose k%2B1} \left[\zeta(k%2B\nu%2B2)-1\right] = 2^{-(\nu%2B1)}

and

\sum_{k=0}^\infty (-1)^k {k%2B\nu%2B1 \choose k%2B2} \left[\zeta(k%2B\nu%2B2)-1\right] = \nu \left[\zeta(\nu%2B1)-1\right] - 2^{-\nu}

and

\sum_{k=0}^\infty (-1)^k {k%2B\nu%2B1 \choose k} \left[\zeta(k%2B\nu%2B2)-1\right] = \zeta(\nu%2B2)-1 - 2^{-(\nu%2B2)}

For integer n\geq 0, the series

S_n = \sum_{k=0}^\infty {k%2Bn \choose k} \left[\zeta(k%2Bn%2B2)-1\right]

can be written as the finite sum

S_n=(-1)^n\left[1%2B\sum_{k=1}^n \zeta(k%2B1) \right]

The above follows from the simple recursion relation S_n%2BS_{n%2B1} = \zeta(n%2B2). Next, the series

T_n = \sum_{k=0}^\infty {k%2Bn-1 \choose k} \left[\zeta(k%2Bn%2B2)-1\right]

may be written as

T_n=(-1)^{n%2B1}\left[n%2B1-\zeta(2)%2B\sum_{k=1}^{n-1} (-1)^k (n-k) \zeta(k%2B1) \right]

for integer n\geq 1. The above follows from the identity T_n%2BT_{n%2B1} = S_n. This process may be applied recursively to obtain finite series for general expressions of the form

\sum_{k=0}^\infty {k%2Bn-m \choose k} \left[\zeta(k%2Bn%2B2)-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%2Bn%2B2)-1}{2^k} {{n%2Bk%2B1} \choose {n%2B1}}=\left(2^{n%2B2}-1\right)\zeta(n%2B2)-1

Expressions in the form of p-series

Adamchik and Srivastava give

\sum_{n=2}^\infty n^m \left[\zeta(n)-1\right] = 1\, %2B \sum_{k=1}^m k!\; S(m%2B1,k%2B1) \zeta(k%2B1)

and

\sum_{n=2}^\infty (-1)^n n^m \left[\zeta(n)-1\right] = -1\, %2B\, \frac {1-2^{m%2B1}}{m%2B1} B_{m%2B1} \,- \sum_{k=1}^m (-1)^k k!\; S(m%2B1,k%2B1) \zeta(k%2B1)

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:

References