The Prime Number Theorem
We return to the question raised back in the
distribution of primes:
roughly how many primes are there below x? The Prime Number
Theorem (PNT) is the precise answer — Gauss's teenage guess, finally proven a century later
with the analytic machinery of the
zeta function.
The statement
The prime-counting function \pi(x) satisfies
\pi(x) \sim \frac{x}{\ln x}, \qquad\text{meaning}\qquad \lim_{x\to\infty} \frac{\pi(x)}{x/\ln x} = 1.
Equivalently, the n-th prime is about n \ln n,
and a random number near x is prime with probability about
1/\ln x. An even sharper estimate uses the logarithmic integral
\operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t}, which hugs
\pi(x) remarkably closely.
How it was proven
Proven independently by Hadamard and de la Vallée Poussin in 1896, the
argument turns entirely on the
zeros of zeta.
The crucial step is showing
\zeta(s) \ne 0 \quad\text{on the line}\quad \Re(s) = 1.
A zero on that line would inject a non-vanishing oscillation into the prime count, breaking the
asymptotic; ruling it out forces \pi(x) to follow
x/\ln x. The primes' large-scale regularity is, quite literally, a
statement about where \zeta cannot be zero. (A century on, "elementary"
proofs by Erdős and Selberg exist too — but the analytic route remains the most illuminating.)
What's left to know
The PNT tells us the main term. The error term — how far
\pi(x) strays from \operatorname{Li}(x) — depends
on how far left the zeta zeros sit. Squeezing that error to its conjectured minimum is
exactly the content of the
Riemann Hypothesis.