John Carlos Baez @johncarlosbaez@mathstodon.xyz
Public

There are fascinating connections between the Riemann zeta function and music theory. I'll probably write a paper about this, but I can't resist talking about a little piece of the story now, as I'm still figuring it out.

Any commutative ring has a zeta function! The Riemann zeta function is the zeta function of ℤ, but the zeta function of ℤ/3 × ℤ/5 is simpler: it's just

1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)

Let's graph this along the 'critical line' where the famous zeros of the Riemann zeta function live. So, let's take

s = ½ + ix

and plot

|1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)|

as a function of x from x = 0 to x = 100. We get this picture here:

(1/n)

Graph of |1/(1 - 3⁻ˢ)(1 - 5⁻ˢ)| as a function of x from x = 0 to x = 100. It's a wiggly blue curve going between about 0.1 and 1, showing vibrations of various different frequencies: fast wiggles and slower 'beats'.
0
0
0

If you have a fediverse account, you can quote this note from your own instance. Search https://mathstodon.xyz/users/johncarlosbaez/statuses/116082085173571063 on your instance and quote it. (Note that quoting is not supported in Mastodon.)