@johncarlosbaez I wonder if there's some model theoretic universality result that all finite (or not too big) commutative rings embed into one specific commutative ring that involves numbers, like the ring of rationals or something like that?

@UweHalfHand - If I didn't have good theoretical/intuitive reasons to expect these beats I would be more suspicious of what I'm seeing. I am suspicious of the height and especially the *width* of the spikes in the Fourier transform of the absolute value of this zeta function - they could easily be delta functions. But I'm not surprised to be getting spikes at frequencies of the form ln(3ʲ5ᵏ)/2π), because if we omit the absolute value, the Fourier transform of the zeta function itself is a weighted sum of delta functions at all possible numbers of the form ln(3ʲ5ᵏ)/2π).

Still, your point is a good one: if I ever want to publish anything about this I should check it more carefully.

The slow 'beats' in the absolute value of the zeta function

1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)

arise because it's the product of two functions: 1/(1 - 3⁻ˢ) which has period

2π/ln(3) ≈ 5.719

as a function of x (remember s = ½ + ix), and 1/(1 - 5⁻ˢ) which has period

2π/ln(5) ≈ 3.904

as a function of x.

These functions both become big at the same point x when there are integers m,n with

x ≈ m 2π/ln(3) ≈ n 2π/ln(5)

This requires

m/n ≈ ln(3)/ln(5)

So finding the tallest peaks in the absolute value of the zeta function amounts to looking for good rational approximations of this number:

ln(3)/ln(5) ≈ 0.682606194

Here are the first few:

2/3 ≈ 0.6667
13/19 ≈ 0.6842
15/22 ≈ 0.6818

The first says we expect tall peaks spaced apart by roughly

2 × [2π/ln(3)] ≈ 3 × [2π/ln(5)]

These numbers are close but not equal! Look at them:

4π/ln(3) ≈ 11.44
6π/ln(5) ≈ 11.71

This explains why I empirically found that the tall sharp spikes in the graph below are separated by a distance of about 11.58. I don't know a formula yet that gives 11.58, but I see why it's between those two numbers!

(4/n)