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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d401704a10ea6c5053b45a41a797051e17169bc2)
where each qn 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71f73de023bc4ba9219365f4abfdae1666e8b82b)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8ac32ace42c1abb5d18d0ec537d75177e7dd54)
and
![{\displaystyle 1-\gamma =\sum _{n=2}^{\infty }{\frac {1}{n}}\left[\zeta (n)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78d15f510077a4bb6119bdb50406ba2475931d3a)
where γ is the Euler–Mascheroni constant. The series
![{\displaystyle \log 2=\sum _{n=1}^{\infty }{\frac {1}{n}}\left[\zeta (2n)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69e98e1cb1a0b8b6e0fceabae103d98cb50f1abe)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b119987227f738ef92beb8b1031b5d95b9daf148)
and
![{\displaystyle {\frac {13}{30}}-{\frac {\pi }{8}}=\sum _{n=1}^{\infty }{\frac {1}{4^{2n}}}\left[\zeta (2n)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675a8c2ed26e324e5c6af200815d844b4b8fd0ec)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88eec406cb0d9554da544b9614fe5581fb6c0786)
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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb165a7307f66782e9907abda770925935cbb6fc)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1388fee99bab58fee441a08d85b8baa60ba0c8)
A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is
.
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d810f56554ac8ae1d82810ffe02f88357844b63)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80e6bbc24b257d55d40c9355ca7a84304071001c)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87800000f217771b53df62a176612c80febf598d)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16f2fb2ee7e0568c9a2b8badbb0932865d12809d)
and
![{\displaystyle \sum _{k=0}^{\infty }(-1)^{k}{k+\nu +1 \choose k+1}\left[\zeta (k+\nu +2)-1\right]=2^{-(\nu +1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5d3c969127771d6343b29fda8c68e1ef01fefa)
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f083804addcac40e42af2534b693d62a606210d)
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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e48542bc883438b81ad6d57f5ba7c62ec35d8b)
For integer n ≥ 0, the series
![{\displaystyle S_{n}=\sum _{k=0}^{\infty }{k+n \choose k}\left[\zeta (k+n+2)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d7aaadfd39daf83925b0325e44ff559ca1c13a)
can be written as the finite sum
![{\displaystyle S_{n}=(-1)^{n}\left[1+\sum _{k=1}^{n}\zeta (k+1)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e171fa93cda2e811c2439c511c9e3150cdf9c512)
The above follows from the simple recursion relation Sn + Sn + 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74b6e438262269f61a65d3d697d748ee8961b114)
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61649b3d0e8d4a90843db44d27274d70dc682313)
for integer n ≥ 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
![{\displaystyle \sum _{k=0}^{\infty }{k+n-m \choose k}\left[\zeta (k+n+2)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/207c86e326ebc5c282f11545fe76c7d263421a9e)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5386bb26f759c1f1d9edd4c418e893f723a9c6)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3bd1e1f0c09e52ca13d4c943ca97d115589223f)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf397c9916184c5f68f78dcf4fc6298aab19f5ab)
where
are the Bernoulli numbers and
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" (PDF). J. Comput. Appl. Math. 121 (1–2): 247–296. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8.
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: CS1 maint: multiple names: authors list (link) - 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.