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 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 For m=2, a number of interesting numbers have a simple expression as rational zeta series: and where γ is the Euler–Mascheroni constant. The series follows by summing the Gauss–Kuzmin distribution. There are also series for π: and 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 series

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: and and and 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 and where B_k are the Bernoulli numbers and S(m,k) are the Stirling numbers of the second kind.

Other series

Other constants that have notable rational zeta series are: