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The generating function for the terms of the harmonic series:

$\frac{1}{n}$

is $-\ln(1 - x)$.

Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ for any $ s > 1$?

That is, does there exist any function $f(x)$ that can be expressed in terms of elementary functions such that $f(x) = \sum_{n=1}^\infty \frac{1}{n^s}x^n$ for some $s > 1$?

I'm assuming that such a function in fact does not exist. Can this be proven to be the case?

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2 Answers 2

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This is an answer to the initial question (asking for a generating function of $\zeta(s)$) :

  • $−\ln(1−x)$ is the generating function of $\ \frac 1n$.
  • $-\frac{\ln(1−x)}{1-x}$ is the generating function of the harmonic number $\ H_n=\sum_{k=1}^n\frac 1k$

The generating function of the generalized harmonic number $\ H_{n,s}:=\sum_{k=1}^n\frac 1{k^s}\ $ is given by : $$\frac{\operatorname{Li}_s(x)}{1-x}$$ with $\operatorname{Li}_s$ the polylogarithm.

Should you simply want $\ \displaystyle\sum_{n=1}^\infty \frac{x^n}{n^s}=\operatorname{Li}_s(x)\ $ then dot dot's answer is right of course !

Note that a generating function for $\zeta(n)$ is known as the digamma function : $$\psi(1+x)=-\gamma-\sum_{n=1}^\infty \zeta(n+1)\;(-x)^n$$ while the reflection formula allows to get the even values of $\zeta$ directly as : $$\pi\;x\;\cot(\pi\;x)=-2\sum_{n=0}^\infty \zeta(2n)\;x^{2n}$$

A generating function for the polylogarithm was obtained too : $$z\,\Phi(z, 1, 1-x)=\sum_{n=0}^\infty\;\operatorname{Li}_{n+1}(z)\;x^n$$ using the Lerch zeta function $\displaystyle\;\Phi(z, s, \alpha) := \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}$.

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  • $\begingroup$ I've edited my question, I of course meant the generating function for the terms of the harmonic series, not for the harmonic series itself (which is divergent as you say.) $\endgroup$
    – Bitrex
    May 26, 2013 at 15:56
  • $\begingroup$ @Bitrex: So that you wanted the generating function of $n^{-s}$ in fact... Note that a generating function for $\zeta(n)$ is known : the digamma function and that a generating function for the even values of $\zeta$ is simply $\pi\cot(\pi\;x)$. $\endgroup$ May 26, 2013 at 16:05
  • $\begingroup$ Nice illustration and summary. $\endgroup$ May 26, 2013 at 16:36
  • $\begingroup$ Thanks @Mhenni ! I must admit that this was done too for my convenience : I searched these generating functions and links more than once! $\endgroup$ May 26, 2013 at 16:47
  • $\begingroup$ And what about The generating function of the polylogarithm.? Or the generatign funciton for the generalised harmonic harmonic numbers BUT in powers of the order, instead of powers of its number of terms? $\endgroup$
    – Dr Potato
    Jun 29, 2021 at 4:25
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The function you are looking for is the Li$_s(z)$, the polylogarithm.

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  • $\begingroup$ Thanks for the reply. It looks like the polylogarithm can only be expressed in terms of elementary functions for integers $s < 1$, so for $s > 1$ it seems the answer to my question is indeed negative. $\endgroup$
    – Bitrex
    May 26, 2013 at 15:32

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