Why the Riemann Hypothesis?

Way back when, I had a post about explaining the Riemann hypothesis in elementary terms. I thought I’d go into some more detail.

The Riemann hypothesis is regarded as one of the outstanding open problems in mathematics. Part of the reason is that it has a certain mystique, since Riemann conjectured it back in 1859, and it has withstood many attempts to prove it since then. A bigger reason is that it solution (either positive or negative) is the main obstacle to answering the question “How many primes are there?”

The fact that there are infinitely many primes goes back to Euclid. The next most logical question is to ask how many primes there are less than a given number. Thanks to the Prime Number Theorem, we know that there are approximately n / ln n primes less than a given number. But this is only an approximation. How good or bad of an approximation is it? We don’t know. That is the question the Riemann hypothesis is trying to answer.