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I am trying to evaluate the following sum:

$\sum_{p = p_0}^{\infty} \frac{x^p}{p^{3/2}} $

where $p_0$ is some integer larger than one and $x$ is smaller than one.

Sums like $\sum_{p = p_0}^{\infty} px^p$ or $\sum_{p = p_0}^{\infty} \frac{x^p}{p} $ can be evaluated by switching sums and integrals, but I don't know how to deal with the $p^{3/2}$. Can anyone help me?

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    @Dejan Govc thanks, that helps a lot. Oh, and actually I think $\zeta(3/2)$ is a quite nice form.2012-01-24

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The Lerch transcendent is defined as

$\Phi(z,s,a)=\sum_{k=0}^\infty \frac{z^k}{(k+a)^s}$

Now,

$\begin{align*} \sum_{p=p_0}^\infty \frac{x^p}{p^{3/2}}&=\sum_{p-p_0=0}^\infty \frac{x^p}{p^{3/2}}\\ &=\sum_{k=0}^\infty \frac{x^k x^{p_0}}{(k+p_0)^{3/2}}\\ &=x^{p_0}\sum_{k=0}^\infty \frac{x^k}{(k+p_0)^{3/2}}=x^{p_0}\Phi\left(x,\frac32,p_0\right) \end{align*}$

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    ... you're right, I was looking for elementary functions. Guess I'm out of luck, yes.2012-01-27