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:

 

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 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:

 

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 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

 

and

 

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:

ReferencesEdit

  • 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 10.1.1.127.9800. doi:10.1524/anly.1998.18.2.131.