# Zeta sum $\sum_{n=2}^\infty \frac{\zeta(n)}{a^n}$

Probably this is known, but mathworld and wolfram alpha don't recognize this potential identities.

Numerical evidence suggests:

$$\sum_{n=2}^\infty \frac{\zeta(n)}{a^n} =? \sum_{n=1}^\infty \frac{1}{a (a n^2 -n) } = \frac{-\gamma - \psi((a-1)/a)}{a} \qquad (1)$$

and

$$\sum_{n=2}^\infty \frac{(-1)^n \zeta(n)}{a^n} =? \sum_{n=1}^\infty \frac{1}{a (a n^2 + n) } = \frac{\gamma + \psi(1/a)+a}{a}$$

for $a \in \mathbb{C}$ whenever the first sum converges.

Web search found the special case $a=2$ is proved in a paper.

Are these identities true?

mathworld 52 gives related identity for even zeta:

$$\sum_{n=0}^\infty\zeta(2n+2) x^{2n}=\frac{1-\pi x \cot(\pi x)}{2 x^2} \qquad (52)$$

Setting $a=\pi$ in (1) and $x=1/\pi$ in (52) gives: $$\sum_{n=1}^\infty\zeta(2n+1)/\pi^{2n+1} = (-\gamma -\psi((\pi-1)/\pi)/\pi - (1-\cot(1))/2$$

Verified with mpmath.nsum and $1000$ digits of precision.

• The first two formulas (question marks) are easy to prove by expanding the Riemann zeta-function into its definition, changing the summation order and using the formula for a geometric series. – Johan Andersson Jun 12 '13 at 9:16
• But of course absolute convergence must be checked to reverse the order of summation. Not hard. – Gerald Edgar Jun 12 '13 at 13:30

Certainly $$\psi(1+z)=−\gamma+\sum_{k=2}^\infty(−1)^k\zeta(k)z^{k−1}$$ and $$\psi(z)=−\gamma−\frac1z+\sum_{k=1}^\infty\frac z{k(k+z)}$$ are classical - see e. g. DLMF

As for Mathematica, it answers to

Sum[Zeta[n]/a^n, {n, 2, \[Infinity]}]


with

(-EulerGamma - PolyGamma[0, 1 - 1/a])/a


to

Sum[((-1)^n Zeta[n])/a^n, {n, 2, \[Infinity]}]


with

(EulerGamma + PolyGamma[0, 1 + 1/a])/a


to

Sum[1/(a (a n^2 - n)), {n, \[Infinity]}]


with

-((EulerGamma + PolyGamma[0, 1 - 1/a])/a)


and to

Sum[1/(a (a n^2 + n)), {n, \[Infinity]}]


with

(EulerGamma + PolyGamma[0, 1 + 1/a])/a