You all know what the natural log function looks like. Take the number 1, divide it by the natural log, and then find the antiderivative of that function. You’ll get the logarithmic integral function. It looks like this:
Sometimes the lower limit of the integral is changed to 2 instead of 0, in order to get rid of that singularity at the origin. But we’re only interested in the behavior at large x so it doesn’t matter either way.
It turns out that if you count all the prime numbers up to one million, the answer will be approximately li(1000000). This is helpful because it’s a lot easier to calculate the logarithmic integral for huge numbers than it is to actually count up all the primes below that number. The formal statement of this fact is called the Prime Number Theorem, and it’s one of the most important facts in mathematics. Here it is:
Pi(x) is not the number pi times x, here pi(x) is actually the prime counting function, which is the number of primes less than x. As x gets larger and larger, pi(x) and the logarithmic integral become closer and closer approximations of each other in terms of their percentage difference.
But that’s in terms of their percent difference. Their absolute difference does grow without bound, it just does so more slowly than x. Now, prove the following statement and you’ll get a million dollars from the Clay Mathematics Institute, the Fields Medal, and a cushy job touring the mathematics lecture circuit. For some particular finite unchanging positive constant c:
That is, the error of the approximation grows more slowly than c times sqrt(x)*ln(x). You don’t even have to figure out the particular value of c, just prove that there is one. And furthermore, it doesn’t even have to be true for all x; just for all x greater than some particular finite positive constant. And you don’t even have to calculate the exact value of that constant either, just prove that there is one.
That statement is a consequence of the Riemann hypothesis, and the truth or falsehood of that statement has been proven to be the same as the truth or falsehood of the Riemann hypothesis. Unfortunately no one has yet been able to prove either statement in either direction despite its being the most important unsolved problem in mathematics.
Hey, it’s the weekend and you’ve probably got some free time. Your million dollars await!