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

:$x=\sum_^\infty q_n \zeta (n,m)$

where each ''q''_''n'' 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

:x=\sum_^\infty q_n \left zeta(n)- \sum_^ k^\right

For ''m=2'', a number of interesting numbers have a simple expression as rational zeta series:

:1=\sum_^\infty \left zeta(n)-1\right/math>

and
:1-\gamma=\sum_^\infty \frac\left zeta(n)-1\right/math>

where γ is the Euler–Mascheroni constant. The series
:\log 2 =\sum_^\infty \frac\left zeta(2n)-1\right/math>

follows by summing the Gauss–Kuzmin distribution. There are also series for π:

:\log \pi =\sum_^\infty \frac\left zeta(n)-1\right/math>

and

:\frac - \frac =\sum_^\infty \frac\left zeta(2n)-1\right/math>

being notable because of its fast convergence. This last series follows from the general identity

:\sum_^\infty (-1)^ t^ \left zeta(2n)-1\right=
\frac + \frac - \frac

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

:$\frac = \sum_^\infty B_n \frac$

Adamchik and Srivastava give a similar series

:$\sum_^\infty \frac \zeta(2n) = \log \left(\frac \right)$

Polygamma-related series

A number of additional relationships can be derived from the Taylor series for the polygamma function at ''z'' = 1, which is
:$\psi^(z+1)= \sum_^\infty (-1)^ (m+k)!\; \zeta (m+k+1)\; \frac$.
The above converges for , ''z'',  < 1. A special case is

:\sum_^\infty t^n \left zeta(n)-1\right=
-t\left gamma +\psi(1-t) -\frac\right

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:

:\sum_^\infty \left zeta(k+\nu+2)-1\right
= \zeta(\nu+2)

where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta

:$\zeta(s,x+y) = \sum_^\infty (-y)^k \zeta (s+k,x)$
taken at ''y'' = −1. Similar series may be obtained by simple algebra:

:\sum_^\infty \left zeta(k+\nu+2)-1\right
= 1

and

:\sum_^\infty (-1)^k \left zeta(k+\nu+2)-1\right
= 2^

and

:\sum_^\infty (-1)^k \left zeta(k+\nu+2)-1\right
= \nu \left zeta(\nu+1)-1\right- 2^

and

:\sum_^\infty (-1)^k \left zeta(k+\nu+2)-1\right
= \zeta(\nu+2)-1 - 2^

For integer ''n'' ≥ 0, the series
:S_n = \sum_^\infty \left zeta(k+n+2)-1\right/math>

can be written as the finite sum

:S_n=(-1)^n\left +\sum_^n \zeta(k+1) \right

The above follows from the simple recursion relation ''S''_''n'' + ''S''_''n'' + 1 = ζ(''n'' + 2). Next, the series

:T_n = \sum_^\infty \left zeta(k+n+2)-1\right/math>

may be written as

:T_n=(-1)^\left +1-\zeta(2)+\sum_^ (-1)^k (n-k) \zeta(k+1) \right

for integer ''n'' ≥ 1. The above follows from the identity ''T''_''n'' + ''T''_''n'' + 1 = ''S''_''n''. This process may be applied recursively to obtain finite series for general expressions of the form

:\sum_^\infty \left zeta(k+n+2)-1\right/math>

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

:$\sum_^\infty \frac =\left(2^-1\right)\left(\zeta(n+2)-1\right)-1$

Expressions in the form of p-series

Adamchik and Srivastava give
:\sum_^\infty n^m \left zeta(n)-1\right=
1\, +
\sum_^m k!\; S(m+1,k+1) \zeta(k+1)

and

:\sum_^\infty (-1)^n n^m \left zeta(n)-1\right=
-1\, +\, \frac B_
\,- \sum_^m (-1)^k k!\; S(m+1,k+1) \zeta(k+1)

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:
* Khinchin's constant
* Apéry's constant

References

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{{Real numbers

Zeta and L-functions
Real numbers