The Riemann Hypothesis

What just happened?

You increased the range N to 10,000. Up close the primes (2, 3, 5, 7, 11, 13, 17…) still look scattered chaotically — there is no simple formula giving the n-th prime. But their overall count, the function $\pi(x)$, grows along a smooth and approximately predictable curve.

The prime-counting function:

Already by the end of the 19th century the prime number theorem had been proved: $\pi(x) \sim \dfrac{x}{\ln x}$ — a simple expression that approximates the seemingly chaotic function well. This is the first trace of a hidden order.

The Riemann zeta function:

In November 1859 Bernhard Riemann published the paper “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” — just six pages that changed number theory. He extended the zeta function, previously known to Euler, to the complex numbers:

and showed that its “zeros” (the places where $\zeta(s)=0$) are closely tied to the exact distribution of the primes.

The hypothesis itself:

Riemann conjectured that all “nontrivial” zeros of the zeta function lie exactly on a single line — the so-called critical line:

To this day no one has proved or disproved it — despite numerical checking for over 10 trillion zeros without a single counterexample. It is one of the 7 Millennium Prize Problems announced by the Clay Mathematics Institute in 2000, with a prize of USD 1 million awaiting a solution.

Why does it matter?

A proof of the Riemann hypothesis would give the most precise possible estimate of the error in the approximation $\pi(x) \approx x/\ln x$ and fundamentally deepen our knowledge of exactly how the primes are distributed — which matters, among other things, for cryptography (the RSA system bases its security on the difficulty of factorization, closely tied to the properties of the primes) and for all of modern number theory.

Why this matters for your exam:

Primes and the Sieve of Eratosthenes are a standard school topic — the Riemann hypothesis shows that behind a seemingly simple notion hides one of the hardest, still-open problems in all of mathematics.

Remember: there are infinitely many primes (Euclid's proof, around 300 BC), but their density among the natural numbers steadily decreases as the range grows — they thin out, even though they never end.