Rational zeta series explained

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 ^{inf ty} _{n =2} 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 ^{inf ty} _{n =2} q _{n} \left[\zeta(n) - \sum ^{m -1} _{k =1} k ^{-n} \right]

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

1= \sum ^{inf ty} _{n =2} \left[\zeta(n) -1 \right]

and

1- \gamma = \sum ^{inf ty} _{n =2} (1)/(n) \left[\zeta(n) -1 \right]

where γ is the Euler–Mascheroni constant. The series

log 2 = \sum ^{inf ty} _{n =1} (1)/(n) \left[\zeta( 2 n) -1 \right]

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

log \pi = \sum ^{inf ty} _{n =2} (2 ( 3 / 2 ) ^{n-3})/(n) \left[\zeta(n) -1 \right]

and

(13)/(30) - (\pi)/(8) = \sum ^{inf ty} _{n =1} (1)/(4 ^{2n}) \left[\zeta( 2 n) -1 \right]

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

\sum ^{inf ty} _{n =1} ( -1 ) ^{n} t ^{2n} \left[\zeta( 2 n) -1 \right] = (t ^{2})/(1+ t ^{2}) + (1- \pi t)/(2) - (\pi t)/(e ^{2\pi} -1)

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

(t)/(e ^{t-1}) = \sum ^{inf ty} _{n =0} B _{n} (t ^{n})/(n!)

Adamchik and Srivastava give a similar series

\sum ^{inf ty} _{n =1} (t ^{2n})/(n) \zeta( 2 n) = log \left( (\pi t)/(\sin (\pi t)) \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 ^{(m)} (z +1 ) = \sum ^{inf ty} _{k =0} ( -1 ) ^{m+k+1} (m + k)! \zeta (m + k +1 ) (z ^{k})/(k!) .The above converges for |z| < 1. A special case is\sum ^{inf ty} _{n =2} t ^{n} \left[\zeta(n) -1 \right] = - t\left[\gamma + \psi( 1- t) - (t)/(1- t) \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 ^{inf ty} _{k =0} {k + \nu +1 \choose k} \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 ^{inf ty} _{k =0} {s + k -1 \choose s -1 } ( - y) ^{k} \zeta (s + k,x) taken at y = -1. Similar series may be obtained by simple algebra:\sum ^{inf ty} _{k =0} {k + \nu +1 \choose k +1 } \left[\zeta(k + \nu +2 ) -1 \right] = 1

and

\sum ^{inf ty} _{k =0} ( -1 ) ^{k} {k + \nu +1 \choose k +1 } \left[\zeta(k + \nu +2 ) -1 \right] = 2 ^{-(\nu+1)}

and

\sum ^{inf ty} _{k =0} ( -1 ) ^{k} {k + \nu +1 \choose k +2 } \left[\zeta(k + \nu +2 ) -1 \right] = \nu \left[\zeta(\nu +1 ) -1 \right] - 2 ^{-\nu} and\sum ^{inf ty} _{k =0} ( -1 ) ^{k} {k + \nu +1 \choose k} \left[\zeta(k + \nu +2 ) -1 \right] = \zeta(\nu +2 ) -1 - 2 ^{-(\nu+2)}

For integer n ≥ 0, the series

S _{n} = \sum ^{inf ty} _{k =0} {k + n \choose k} \left[\zeta(k + n +2 ) -1 \right]

can be written as the finite sum

S ^{n} _{k =1} \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 ^{inf ty} _{k =0} {k + n -1 \choose k} \left[\zeta(k + n +2 ) -1 \right]

may be written as

T ^{n +1} _{n = ( -1 )} \left[n +1- \zeta( 2 ) + \sum ^{n -1} _{k =1} ( -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 ^{inf ty} _{k =0} {k + n - m \choose k} \left[\zeta(k + n +2 ) -1 \right]

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 ^{inf ty} _{k =0} (\zeta(k + n +2 ) -1)/(2 ^{k}) {{n + k +1 } \choose {n +1 }} = \left( 2 ^{n+2} -1 \right)\left(\zeta(n +2 ) -1 \right) -1

Expressions in the form of p-series

Adamchik and Srivastava give

\sum ^{inf ty} _{n =2} n ^{m} \left[\zeta(n) -1 \right] = 1 + \sum ^{m} _{k =1} k! S(m +1 ,k +1 ) \zeta(k +1 )

and

\sum ^{inf ty} _{n =2} ( -1 ) ^{n} n ^{m} \left[\zeta(n) -1 \right] = -1 + (1-2 ^{m+1})/(m +1) B _{m+1} - \sum ^{m} _{k =1} ( -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

• Jonathan M. Borwein, David M. Bradley, Richard E. Crandall. Computational Strategies for the Riemann Zeta Function. J. Comput. Appl. Math.. 2000. 121. 1–2. 247–296. 10.1016/s0377-0427(00)00336-8. 2000JCoAM.121..247B. free.
• Victor S. Adamchik and H. M. Srivastava. Some series of the zeta and related functions. Analysis. 1998. 18. 2. 131–144. 10.1524/anly.1998.18.2.131. 10.1.1.127.9800. 11370668.