Prime Number Theorem

The Prime Number Theorem is one of the crowning achievements of mathematics, connecting prime distribution to complex analysis.

TheoremPrime Number Theorem (Hadamard-de la Vallée Poussin, 1896)

\pi(x) \sim \frac{x}{\log x} \quad \text{as } x \to \infty

Equivalently:

\lim_{x \to \infty} \frac{\pi(x) \log x}{x} = 1

Or in terms of the logarithmic integral:

\pi(x) \sim \text{Li}(x) = \int_2^x \frac{dt}{\log t}

Remark

The Prime Number Theorem is equivalent to each of:

$\psi(x) \sim x$ where $\psi(x) = \sum_{n \leq x} \Lambda(n)$
$\theta(x) \sim x$ where $\theta(x) = \sum_{p \leq x} \log p$
$\sum_{p \leq x} 1/p \sim \log \log x$
$p_n \sim n \log n$ (the $n$ -th prime)

ExampleHeuristic Derivation

If primes were "random" with "probability" $1/\log n$ of $n$ being prime:

\pi(x) \approx \sum_{n \leq x} \frac{1}{\log n} \approx \int_2^x \frac{dn}{\log n} = \text{Li}(x) \approx \frac{x}{\log x}

Remarkably, this naive probabilistic argument gives the correct answer!

DefinitionError Terms

With explicit error, the PNT states:

\pi(x) = \text{Li}(x) + O(x e^{-c\sqrt{\log x}})

Under the Riemann Hypothesis:

\pi(x) = \text{Li}(x) + O(\sqrt{x} \log x)

The error term is directly related to the location of zeros of $\zeta(s)$ .

ExamplePrimes in Short Intervals

PNT implies that for $x^{\theta} \leq y \leq x$ with $\theta > 1/2$ :

\pi(x+y) - \pi(x) \sim \frac{y}{\log x}

The number of primes in $[x, x+y]$ is approximately $y/\log x$ , showing primes have density $1/\log x$ at scale $x$ .

The Prime Number Theorem transforms our understanding of primes from mysterious to statistical, revealing that despite their irregular individual behavior, primes exhibit beautiful regularity in aggregate.