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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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Did you hear about polylogarithms ? en.wikipedia.org/wiki/Polylogarithm –  dot dot May 26 '13 at 15:02

2 Answers 2

up vote 8 down vote accepted

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 : $$\pi\;x\;\cot(\pi\;x)=-2\sum_{n=0}^\infty \zeta(2n)\;x^{2n}$$

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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.) –  Bitrex May 26 '13 at 15:56
    
@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 $\pi\cot(\pi\;x)$. –  Raymond Manzoni May 26 '13 at 16:05
    
Nice illustration and summary. –  Mhenni Benghorbal May 26 '13 at 16:36
    
Thanks @Mhenni ! I must admit that this was done too for my convenience : I searched these generating functions and links more than once! –  Raymond Manzoni May 26 '13 at 16:47

The function you are looking for is the Li$_s(z)$, the polylogarithm.

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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. –  Bitrex May 26 '13 at 15:32

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