Rational zeta series

From Wikipedia, the free encyclopedia

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 ζ(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 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

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

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

$\psi ^{{(m)}}(z+1)=\sum _{{k=0}}^{\infty }(-1)^{{m+k+1}}(m+k)!\;\zeta (m+k+1)\;{\frac {z^{k}}{k!}}$ .

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 S_n + S_n + 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 T_n + T_n + 1 = S_n. This process may be applied recursively to obtain finite series for general expressions of the form

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

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

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:

References

Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". J. Comp. App. Math. 121: p.11.
Victor S. Adamchik and H. M. Srivastava (1998). "Some series of the zeta and related functions". Analysis 18: pp. 131–144.

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.