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 each 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

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

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

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

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 ${\displaystyle B_{k}}$ 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. Comput. Appl. Math. 121 (1–2): 247–296. Bibcode:2000JCoAM.121..247B. 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 (2): 131–144. CiteSeerX 10.1.1.127.9800. doi:10.1524/anly.1998.18.2.131. S2CID 11370668.