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

${\displaystyle x=\sum_{n=2}^\infty q_n \zeta (n,m)}$

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

${\displaystyle x=\sum_{n=2}^\infty q_n \left[\zeta(n)- \sum_{k=1}^{m-1} k^{-n}\right] }$

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

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

and

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

where γ is the Euler–Mascheroni constant. The series

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

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

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

and

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

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

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

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

${\displaystyle \frac{x}{e^x-1} = \sum_{n=0}^\infty B_n \frac{t^n}{n!}}$

Adamchik and Srivastava give a similar series

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

Polygamma-related series

A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is

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

The above converges for |z| < 1. A special case is

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

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:

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

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

${\displaystyle \zeta(s,x+y) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x)}$

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

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

and

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

and

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

and

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

For integer n ≥ 0, the series

${\displaystyle S_n = \sum_{k=0}^\infty {k+n \choose k} \left[\zeta(k+n+2)-1\right]}$

can be written as the finite sum

${\displaystyle S_n=(-1)^n\left[1+\sum_{k=1}^n \zeta(k+1) \right] }$

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

${\displaystyle T_n = \sum_{k=0}^\infty {k+n-1 \choose k} \left[\zeta(k+n+2)-1\right]}$

may be written as

${\displaystyle T_n=(-1)^{n+1}\left[n+1-\zeta(2)+\sum_{k=1}^{n-1} (-1)^k (n-k) \zeta(k+1) \right] }$

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

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

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

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

Expressions in the form of p-series

Adamchik and Srivastava give

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

and

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

where $B_{k}$ are the Bernoulli numbers and ${\displaystyle S(m,k)}$ 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

• Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). J. Comp. App. Math. 121: p.11. doi:10.1016/s0377-0427(00)00336-8.
• Victor S. Adamchik and H. M. Srivastava (1998). "Some series of the zeta and related functions" (PDF). Analysis 18: pp. 131–144. doi:10.1524/anly.1998.18.2.131.