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
where 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.
For integer m>1, one has
For m=2, a number of interesting numbers have a simple expression as rational zeta series:
where γ is the Euler–Mascheroni constant. The series
follows by summing the Gauss–Kuzmin distribution. There are also series for π:
being notable because of its fast convergence. This last series follows from the general identity
Adamchik and Srivastava give a similar series
The above converges for |z| < 1. A special case is
where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta
taken at y = −1. Similar series may be obtained by simple algebra:
For integer n ≥ 0, the series
can be written as the finite sum
The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series
may be written as
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
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
Expressions in the form of p-series
Adamchik and Srivastava give
Other constants that have notable rational zeta series are:
- 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.1918.104.22.168.