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://cdn.statically.io/img/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.
For integer m>1, one has
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For m=2, a number of interesting numbers have a simple expression as rational zeta series:
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and
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where γ is the Euler–Mascheroni constant. The series
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follows by summing the Gauss–Kuzmin distribution. There are also series for π:
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and
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being notable because of its fast convergence. This last series follows from the general identity
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which in turn follows from the generating function for the Bernoulli numbers
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Adamchik and Srivastava give a similar series
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A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is
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The above converges for |z| < 1. A special case is
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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:
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where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta
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taken at y = −1. Similar series may be obtained by simple algebra:
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and
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and
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and
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For integer n ≥ 0, the series
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can be written as the finite sum
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The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series
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may be written as
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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
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for positive integers m.
Half-integer power series
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Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one has
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Adamchik and Srivastava give
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and
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where are the Bernoulli numbers and are the Stirling numbers of the second kind.
Other constants that have notable rational zeta series are: