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(sm) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.

Elementary seriesEdit

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


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


Adamchik and Srivastava give a similar series


Polygamma-related seriesEdit

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


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:


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 seriesEdit

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-seriesEdit

Adamchik and Srivastava give




where   are the Bernoulli numbers and   are the Stirling numbers of the second kind.

Other seriesEdit

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 (1–2): 247–296. doi:10.1016/s0377-0427(00)00336-8.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 doi:10.1524/anly.1998.18.2.131.