Primes thin out at a precise rate that a teenage Gauss first guessed
Among whole numbers up to 1000 digits long, roughly one in 2300 is prime; double the length to 2000 digits and the odds fall to about one in 4600. The prime number theorem pins down exactly how primes grow scarcer, a pattern Carl Friedrich Gauss recalled noticing at 15 or 16.
The theorem concerns the prime-counting function, the number of primes up to a given size; for 10 it equals four, counting 2, 3, 5 and 7. It says this count behaves like x divided by the natural logarithm of x, in the sense that their ratio tends to 1 as x grows. Equivalently, near a large number N the chance that a random integer is prime is about 1 over log N, and the typical gap between neighbouring primes is about log N. The claim is about relative error only; the absolute difference between the two quantities is not controlled.
Another formulation estimates the nth prime as roughly n times log n. Take n as 2 times 10 to the 17th: the true prime is 8512677386048191063, while the estimate rounds to 7967418752291744388, off by about 6.4 percent. The theorem is also equivalent to statements about Chebyshev's functions and about sums of the Möbius function.
The conjectures came from tables. Using lists by Anton Felkel and Jurij Vega, Adrien-Marie Legendre proposed around 1797 or 1798 a formula with two unknown constants, refining it in 1808 to use the value minus 1.08366. Gauss, writing in 1849, said he had pondered the question in 1792 or 1793. In 1838 Peter Gustav Lejeune Dirichlet suggested the logarithmic integral, which proved far more accurate when differences rather than ratios are compared.
Pafnuty Chebyshev attacked the problem in papers of 1848 and 1850, using the zeta function for real arguments as Euler had done in 1737. He showed that if the ratio had a limit, it had to be 1, and trapped it between 0.92129 and 1.10555 for large x, enough to prove Bertrand's postulate that a prime always lies between n and 2n. Bernhard Riemann's 1859 memoir supplied the decisive tools, and in 1896 two mathematicians working separately, Hadamard and de la Vallée Poussin, each completed a proof.
Source: Prime number theorem
