50 Mathematical Ideas You Really Need to Know

50 The Riemann hypothesis

The Riemann hypothesis represents one of the stiffest challenges in pure mathematics.
The Poincaré conjecture and Fermat’s last theorem have been conquered but not the
Riemann hypothesis. Once decided, one way or the other, elusive questions about the
distribution of prime numbers will be settled and a range of new questions will be
opened up for mathematicians to ponder.

The story starts with the addition of fractions of the kind

The answer is 1⅚ (approximately 1.83). But what happens if we keep adding
smaller and smaller fractions, say up to ten of them?

Using only a handheld calculator, these fractions add up to approximately 2.9
in decimals. A table shows how the total grows as more and more terms are
added.
The series of numbers