zeta and theta

G = \prod_{m=2}^{\infty} \frac{1}{\zeta(m)}
We can rewrite this:
\prod_{m=2}^{\infty} \prod_{\mathrm{p prime}} \frac{1}{1-1/p^{m}}
Consider at this point we can swap products:
\prod_{\mathrm{p prime}}  \prod_{m=2}^{\infty} \frac{1}{1-1/p^{m}}
We can then rewrite the inner product:
\prod_{\mathrm{p prime}} \left(\prod_{m=2}^{\infty} 1 - 1/p^{m}\right)^{-1}

Written out fully, we get, after consulting mathworld:

The inner is almost a product of Jacobi theta functions:

\prod_{m=2}^{\infty} \frac{1}{\zeta(m)} = \prod_{\mathrm{p prime}} \frac{p}{p-1} \left(\sqrt[24]{p}\sqrt[3]{\frac{1}{2}\theta_{1}'(0,1/\sqrt{p})}\right)

It is interesting that this is almost, but not quite the product of the probabilities that a large random p-ary matrix is nonsingular, see A048651, and the logical question then becomes, page 27. How does enumerating finite nilpotent groups correspond to the product of the zeta function evaluated at n\geq 2: the question is, what is the more abstract viewpoint on the swapping of the products?

Moreover, it would be interesting to find a good Hecke correspondence angle on this, but I don’t know enough about Hecke correspondence to make the right statement about it.

Additional: I wonder if it is possible to employ the bewilderingly large range of identities for the theta function so that the side with the zeta on it turns into some kind of invariant.


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