Prime Number Theorem

The Prime Number Theorem describes the asymptotic distribution of prime numbers. Let $π (x)$ prime counting function. Then:

π (x) \sim \frac{x}{\ln (x)}

which means:

lim_{x \to + \infty} \frac{π (x)}{\frac{x}{\ln (x)}} = 1

Contributors to the theorem:

Charles Jean de la Vallée-Poussin (1866–1962)
Jacques Hadamard (1865–1963)

Both independently proved the theorem in 1896 using complex analysis and properties of the Riemann zeta function.

A sharper version: integrating the density

The statement $π (x) \sim \frac{x}{\ln x}$ is only a crude way to read the theorem. A much better approximation comes from recalling that primes are distributed with density $δ (t) \approx \frac{1}{\ln t}$ near $t$ (see prime density). Integrating this density over all $t$ up to $x$ gives the expected number of primes:

π (x) \approx \int_{2}^{x} \frac{d t}{\ln t} = Li (x),

the logarithmic integral. This is a far more accurate estimate of $π (x)$ than $\frac{x}{\ln x}$ :

$Li (x)$ tracks $π (x)$ with a relative error that goes to zero faster; e.g. the error $π (x) - Li (x)$ at powers of 10 is orders of magnitude smaller than the error of $\frac{x}{\ln x}$ .
The real content of the prime number theorem is the asymptotic statement $π (x) \sim Li (x)$ , i.e.

lim_{x \to + \infty} \frac{π (x)}{Li (x)} = 1,

and $\frac{x}{\ln x}$ is simply the leading-order approximation to $Li (x)$ , hence to $π (x)$ .

The size of the residual error $| π (x) - Li (x) |$ is known to be $o (x)$ , but the sharp order remains open. The Riemann hypothesis is equivalent to the bound $| π (x) - Li (x) | = O (\sqrt{x} \log x)$ .

What we know for sure about the error

We know:

(lim_{x \to \infty} \frac{| π (x) - Li (x) |}{x} = 0)

and also

lim_{x \to \infty} \frac{| π (x) - Li (x) |}{x^{0.5}} = undefined

So,

Order (| π (x) - Li (x) |) > \sqrt{x}