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Evaluate:

$$S = \sum_{n=1}^{\infty} \frac{H_n}{n^2}$$

Using complex analysis.

I just needs hints, I have no attempts,

but I believe is has to do with residues.

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  • $\begingroup$ Have you seen any similar problems done with residues? $\endgroup$ – Gerry Myerson Jan 16 '15 at 19:57
  • $\begingroup$ I have but not with $H_n$ $\endgroup$ – Amad27 Jan 16 '15 at 19:57
  • $\begingroup$ what is the meaning of $H_{n}$? $\endgroup$ – Mark Hubenthal Jan 17 '15 at 2:03
  • $\begingroup$ @Mark, probably $H_n=\sum_1^n(1/k)$. $\endgroup$ – Gerry Myerson Jan 17 '15 at 3:40
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By using a lemma you already asked to prove, if we take $f(z)=(\gamma+\psi(-z))^2$, the poles of $f(z)$ occur in $z=0,1,2,\ldots$ and for any $n\in\mathbb{N}^*$ we have: $$\operatorname{Res}\left(f(z),z=n\right) = 2H_n,$$ so: $$ \sum_{n=1}^{+\infty}\frac{H_n}{n^2}=-\operatorname{Res}\left(\frac{f(z)}{z^2},z=0\right)=-\operatorname{Res}\left(\left(\frac{\gamma+\psi(-z)}{z}\right)^2,z=0\right).\tag{1}$$ On the other hand, since in a punctured neighbourhood of $z=0$: $$\gamma+\psi(-z) = \frac{1}{z}-\zeta(2) z-\zeta(3)z^2-\zeta(4)z^4+\ldots \tag{2}$$ it happens that: $$\left(\frac{\gamma+\psi(-z)}{z}\right)^2=\frac{1}{z^4}-\frac{2\zeta(2)}{z^2}-\frac{2\zeta(3)}{z}+\frac{\zeta(4)}{2}+\ldots\tag{3}$$ so, by $(1)$, $$ \sum_{n=1}^{+\infty}\frac{H_n}{n^2}=\color{red}{2\,\zeta(3)}.\tag{4}$$

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  • $\begingroup$ Are you using the zeta definition of $\psi(-z) + \gamma$ in $(2)$ ?? Where can I see it? $\endgroup$ – Amad27 Jan 17 '15 at 14:10
  • $\begingroup$ From en.wikipedia.org/wiki/Digamma_function#Taylor_series $$\psi(z+1) + \gamma = -\sum_{n=1}^{\infty} \zeta(n+1)(-z)^n$$ $$\psi(-z) + \gamma = -\sum_{n=1}^{\infty} \zeta(n+1)(z + 1)^n$$ this does give your series. $\endgroup$ – Amad27 Jan 17 '15 at 14:12
  • $\begingroup$ @Amad27, you are asking the same things over and over. The Laurent series in $(2)$ arises from considering the logarithmic derivative of the Weierstrass product for the $\Gamma$ function, or the integral representation of the $\psi$ function. Anyway, I remember you used relation $(2)$ somewhere, so you should be aware of its proof. $\endgroup$ – Jack D'Aurizio Jan 17 '15 at 14:15
  • $\begingroup$ And remember that $\psi(z+1)=\frac{1}{z}+\psi(z)$. $\endgroup$ – Jack D'Aurizio Jan 17 '15 at 14:16
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    $\begingroup$ A punctured plane, Amad, is a plane with a point removed; in this case, the origin is the point that is removed. Do you not have a complex variables text where you can read about these things (instead of expecting Jack to be your private, free, tutor)? or maybe you have an instructor/lecturer/professor you can ask? $\endgroup$ – Gerry Myerson Jan 17 '15 at 19:34

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