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

$x=\sum _{n=2}^{\mathrm{\infty }}{q}_n\zeta (n,m)$

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

$x=\sum _{n=2}^{\mathrm{\infty }}{q}_n[\zeta (n)-\sum _{k=1}^{m-1}{k}^{-n}]$

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

$1=\sum _{n=2}^{\mathrm{\infty }}[\zeta (n)-1]$

and

$1-\gamma =\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n}[\zeta (n)-1]$

where γ is the Euler–Mascheroni constant. The series

$\log 2=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}[\zeta (2n)-1]$

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

$\log \pi =\sum _{n=2}^{\mathrm{\infty }}\frac{2(3/2{)}^n-3}{n}[\zeta (n)-1]$

and

$\frac{13}{30}-\frac{\pi }{8}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{4^{2n}}[\zeta (2n)-1]$

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

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{t}^{2n}[\zeta (2n)-1]=\frac{t^2}{1+t^2}+\frac{1-\pi t}{2}-\frac{\pi t}{e^{2\pi t}-1}$

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

$\frac{t}{e^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\frac{t^n}{n!}$

Adamchik and Srivastava give a similar series

$\sum _{n=1}^{\mathrm{\infty }}\frac{t^{2n}}{n}\zeta (2n)=\log \left(\frac{\pi t}{\sin (\pi t)}\right)$

Polygamma-related series

A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is

${\psi }^{(m)}(z+1)=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{m+k+1}(m+k)!\zeta (m+k+1)\frac{z^k}{k!}$.

The above converges for |z| < 1. A special case is

$\sum _{n=2}^{\mathrm{\infty }}{t}^n[\zeta (n)-1]=-t[\gamma +\psi (1-t)-\frac{t}{1-t}]$

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:

$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+\nu +1}{k})[\zeta (k+\nu +2)-1]=\zeta (\nu +2)$

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

$\zeta (s,x+y)=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{s+k-1}{s-1})(-y{)}^k\zeta (s+k,x)$

taken at y = −1. Similar series may be obtained by simple algebra:

$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+\nu +1}{k+1})[\zeta (k+\nu +2)-1]=1$

and

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(\genfrac{}{}{0ex}{}{k+\nu +1}{k+1})[\zeta (k+\nu +2)-1]=2^{-(\nu +1)}$

and

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(\genfrac{}{}{0ex}{}{k+\nu +1}{k+2})[\zeta (k+\nu +2)-1]=\nu [\zeta (\nu +1)-1]-2^{-\nu }$

and

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k(\genfrac{}{}{0ex}{}{k+\nu +1}{k})[\zeta (k+\nu +2)-1]=\zeta (\nu +2)-1-2^{-(\nu +2)}$

For integer n ≥ 0, the series

$S_n=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+n}{k})[\zeta (k+n+2)-1]$

can be written as the finite sum

$S_n=(-1{)}^n[1+\sum _{k=1}^n\zeta (k+1)]$

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

$T_n=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+n-1}{k})[\zeta (k+n+2)-1]$

may be written as

$T_n=(-1{)}^{n+1}[n+1-\zeta (2)+\sum _{k=1}^{n-1}(-1{)}^k(n-k)\zeta (k+1)]$

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

$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{k+n-m}{k})[\zeta (k+n+2)-1]$

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

$\sum _{k=0}^{\mathrm{\infty }}\frac{\zeta (k+n+2)-1}{2^k}(\genfrac{}{}{0ex}{}{n+k+1}{n+1})=(2^{n+2}-1)(\zeta (n+2)-1)-1$

Expressions in the form of p-series

Adamchik and Srivastava give

$\sum _{n=2}^{\mathrm{\infty }}{n}^m[\zeta (n)-1]=1+\sum _{k=1}^{m}k!S(m+1,k+1)\zeta (k+1)$

and

$\sum _{n=2}^{\mathrm{\infty }}(-1{)}^n{n}^m[\zeta (n)-1]=-1+\frac{1-2^{m+1}}{m+1}{B}_{m+1}-\sum _{k=1}^m(-1{)}^{k}k!S(m+1,k+1)\zeta (k+1)$

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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