Voronin's Universality Theorem

[usernameguy]

The Riemann Zeta function is a relatively famous mathematical function that has a number of remarkable properties.

One of the most remarkable was discovered by the Russian mathematician Voronin in 1975: its "universality".

In short, the Riemann Zeta function has within it all other functions. That is, along a particular strip of the function, you can find, in miniature, a piece of any other function you can name (so long as it's continuous & analytic). This includes simple parabolas, more complex and chaotic shapes, a relief of the Earth's surface, and a relief of Mickey Mouse.

This may not sound like a lot. But consider that you can can translate and dilate words into a 2-d area that would fit on the special strip of the Riemann Zeta function. So you can map areas to words and words to areas.

Thus, anything you can name appears in the Riemann Zeta. Anything at all:

• the Declaration of Independence
• that picture your parents took of you when you were 4 (encoded as a JPEG then uuencoded)
• this post
• the current state of the Universe

To me, this says more about the nature of infinity than anything else. The Riemann Zeta function is infinitely complex. It has infinite entropy. We tend to think about infinite functions as if they're merely very big. On the special strip in the Riemann Zeta function, you can scan and find anything at all if you're just patient enough*.

But something truly infinite can do things we have trouble intuitively understanding. It's the Riemann Zeta's combination of intricacy and its true infinity that allows it to do this.

For a more technical discussion of this, see this and this.

* (That said, note there are things that are literally impossible to find on it. If you took the largest, most efficient computer you could construct, and ran it for the full lifetime of the Universe, there are still things outside that scan area. An infinite number of things, in fact. Infinity is bigger than that.)