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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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Asking for the general evaluation is probably too optimistic - even with $x=1$ and $p_0=1$ the sum is $\zeta (1/2) $ which has no nicer form. –  Ragib Zaman Jan 24 '12 at 11:04
    
@Ragib, don't you mean $\zeta(3/2)$? But your point is absolutely right, there's no reason to expect an evaluation in terms of the usual functions. –  Gerry Myerson Jan 24 '12 at 11:27
    
@GerryMyerson Sorry, you are correct, that is what I meant. –  Ragib Zaman Jan 24 '12 at 11:49
    
But of course, we could just give it a name "the john_leo" function and the question would be simple. –  Fabian Jan 24 '12 at 11:52
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Actually, this seems to have been studied before. I plugged it into Wolfram|Alpha and it says this is basically the Lerch Transcendent. –  Dejan Govc Jan 24 '12 at 12:51

1 Answer 1

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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Yes, I've understood that. I was trying to look for a closed form because I need it for a function asymptotic. Thanks though –  john_leo Jan 27 '12 at 11:35
    
...the Lerch transcendent is a closed form. If what you wanted was an expression in terms of elementary functions, you're out of luck. –  J. M. Jan 27 '12 at 11:51
    
... you're right, I was looking for elementary functions. Guess I'm out of luck, yes. –  john_leo Jan 27 '12 at 11:54

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