The Riemann Hypothesis for Dummies; or, How Bad Music can Make You Obscenely Rich

No, this blog post isn't about 50 Cent.

Here's the scenario:

I'm an ultra-modern, avant-garde composer. I...compose music. One day, I decide, "Hey! I'm gonna write the weirdest piece EVER!"

Here's what I do:

For strings, I write a waltz (which, as you probably know, is in 3/4 time.) But actually, I lied. I don't write a waltz; I only write one measure of a waltz. Which seems like a weird thing to do, but you'll see my plan.

Then, for horns, I write, say, one measure of 4/4 time. And for piano, one measure of, let's say, 7/4 time. And so on.

Then I instruct each section to play their one single measure over and over and over again until they get tired, and then to keep playing it until all the sections finish their respective measures at the same moment.

The musicians will gripe about this arrangement, as musicians often do, but the overall sound won't repeat for a very long time! For example: If, as above, the strings play in 3/4, the horns in 4/4, and the piano in 7/4, then the piece will last a total of 3*4*7 = 84 beats.

But there's a catch! Yes, I'm an edgy and avant-garde composer, but I'm also environmentally conscious. Which means I want to make this piece as long as I possibly can (without repeating itself, of course) while only using a fixed number of beats, so that I won't waste paper. (Don't think too hard about that part. It's avant-garde composer logic.)

So, here's my question: Exactly how long can I make my piece last without writing more than, oh, n total beats?

Believe it or not, this question is actually unsolved! More than 100 years ago, in 1902, a German mathematician named Edmund Landau proved that when n is really big, I can't make it go on for much longer than e^\sqrt{n ln n} beats (where e is the base of the natural logarithms, or 2.71828...) This is a weird function! It grows a LOT slower than any exponential function (e.g. 1,2,4,8,...) but a LOT faster than any polynomial (for example, n^2, or 1,4,9,16...).

But mathematicians think they can do better than Landau. In fact, they think that something similar to Landau's answer -- exactly the same, in fact, except that n ln n has been replaced by something called "Li^-1(n)," which behaves very similarly -- provides an upper bound for big enough n, not just a close-to-upper bound.

But no one's been able to prove their conjecture! In fact, mathematicians have shown that this conjecture is precisely equivalent to the famous Riemann Hypothesis.

And the RH is considered to be one of the hardest problems in mathematics. It's so hard, in fact, that the Clay Mathematics Institute is offering a million-dollar reward to the first person who can publish a correct proof. In other words, my composer's idle question is actually worth seven figures.