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Riemann's Xi-function is defined as

$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$

At the same time we have the following formulas for n-sphere's area and volume:

$$\begin{array}{ll} S_{n}(R) &= \displaystyle{\frac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}R^{n-1}} \\[1 em] V_n(R) &= \displaystyle{\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}}R^n \end{array}$$

It seems that these formulas are similar. Does Riemann's Xi function provide a sort of connection between n-spheres of positive and negative dimension?

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    $\begingroup$ What do you mean by a sphere of negative dimension? I guess you know that the volume computation is fairly elementary, by integrating $\exp(-r^2/2)$ for $r = |x|$ over $\mathbb{R}^n$ in cartesian and polar coordinates and comparing the results. $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2015 at 17:18
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    $\begingroup$ @ToddTrimble I think the OP is wondering whether there is any philosophical significance to the formal similarity of the forms, not how to compute them (though I could be wrong) $\endgroup$
    – Igor Rivin
    Commented Oct 25, 2015 at 21:25
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    $\begingroup$ @IgorRivin My question was directed at "spheres of negative dimension", and not so much how to compute. But leaving aside spheres of negative dimension, it's reasonable to wonder at that similarity. My lay understanding is that the $\pi^{-s/2} \Gamma(\frac{s}{2})$ is interpretable as the Euler factor at the archimedean place, analogous to $1/(1-p^{-s})$ at a non-archimedean place $p$, and that all that comes under the rubric "Tate's thesis" provides a conceptual explanation of this. It seems not unreasonable to surmise a connection with the area computation. $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2015 at 21:43

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