Statement of Prime Number Theorem

The Prime Number Theorem (PNT) is the crown jewel of 19th century mathematics, settling the distribution of primes with remarkable precision.

TheoremPrime Number Theorem

As $x \to \infty$ :

\pi(x) \sim \frac{x}{\log x}

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

Equivalently: $\psi(x) \sim x$ and $\theta(x) \sim x$ .

ExampleNumerical Evidence

| $x$ | $\pi(x)$ | $x/\log x$ | Ratio | |-----|----------|------------|-------| | $10^3$ | 168 | 145 | 1.159 | | $10^6$ | 78,498 | 72,382 | 1.084 | | $10^9$ | 50,847,534 | 48,254,942 | 1.054 | | $10^{12}$ | 37,607,912,018 | 36,191,206,825 | 1.039 |

The ratio approaches 1, confirming PNT. Using $\text{Li}(x)$ gives even better approximations.

DefinitionEquivalent Formulations

The following are equivalent to PNT:

$\pi(x) \sim x/\log x$
$\psi(x) \sim x$
$\theta(x) \sim x$
$\sum_{p \leq x} \frac{1}{p} \sim \log \log x$ (Mertens' theorem)
$\prod_{p \leq x} \left(1 - \frac{1}{p}\right)^{-1} \sim \log x$
The $n$ -th prime satisfies $p_n \sim n \log n$

Remark

The PNT was conjectured by Gauss and Legendre around 1800 based on numerical tables. It was independently proved by Hadamard and de la Vallée Poussin in 1896 using complex analysis, specifically the zero-free region of $\zeta(s)$ on the line $\Re(s) = 1$ .

Elementary proofs (avoiding complex analysis) were later found by Erdős and Selberg in 1949, though they are significantly longer and less conceptual.

ExamplePrime Number Theorem with Error Term

With explicit constants:

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

for some $c > 0$ . Under the Riemann Hypothesis:

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

which is optimal up to logarithmic factors.

The Prime Number Theorem transforms primes from a mysterious, irregular sequence into a well-understood statistical ensemble, paving the way for modern analytic number theory.