# product of ζ(n) over N\1 and random p-ary matrices.

Let’s take a look at:

$\prod_{n=2}^{\infty} \zeta(n)$

Where $\zeta(n)$ is the Riemann zeta function of $n$. Numerically this is around
2.2948565916733137, but that’s not so interesting. What’s interesting is what happens when
we write out the Euler product for $\zeta(n)$:

$\prod_{n=2}^{\infty} \zeta(n) = \prod_{n=2}^{\infty} \prod_{p \mathrm{prime}} \frac{1}{1-p^{-n}}$

$\prod_{p \,\mathrm{prime}} \prod_{n=2}^{\infty} \frac{1}{1-p^{-n}} = \prod_{p \,\mathrm{prime}} \left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right]^{-1}$

So maybe we want this instead:

$\prod_{n=2}^{\infty} \frac{1}{\zeta(n)} = \prod_{p \,\mathrm{prime}} \left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right] =$

$\prod_{p \,\mathrm{prime}} \frac{p}{p-1} \left[\prod_{n=1}^{\infty} 1-\frac{1}{p^{n}}\right] \approx 0.43575707677264559373\ldots$

What’s interesting about this is that $\left[\prod_{n=1}^{\infty} 1-\frac{1}{2^{n}}\right]$ corresponds to oeis:A048651 — “This is the probability that a large random binary matrix is nonsingular ”, and it’s not hard to see that this generalizes to large random $p$-ary matrices.

Remarks:
1. There is some connection between the nontrivial zeros of $\zeta(s)$ and random matrices, the Gaussian Unitary Ensemble (GUE) in particular. But it is interesting that we have gotten to random matrices without even considering the zeroes.
2. The product $\left[\prod_{n=2}^{\infty} 1-\frac{1}{p^{n}}\right]$ can be expressed in terms of a Jacobian theta functions or a q-Pochhammer symbol. It is instructive to note that we have gotten to the territory of modular forms without having to deal with Hecke’s generalization of the identity Riemann used to construct the zeta function from theta functions.
3. The constant $0.435757076772645593\ldots$ is mentioned in Steven Finch‘s Errata and Addenda to Mathematical Constants on page 30 where he says of it “makes a small appearence (as a certain
best probability corresponding to finite nilpotent groups)”