The Prime Number Theorem

The Prime Number Theorem (PNT) is a cornerstone of number theory that provides an approximate description of how prime numbers are distributed. The prime-counting function, \(\pi(x)\), is a step function that jumps by 1 at each prime number. While the exact location of primes appears random, the PNT reveals a regular asymptotic behavior. The theorem doesn’t say that the difference between \(\pi(x)\) and \(x/\ln(x)\) is small, but rather that their ratio approaches 1 as \(x\) becomes arbitrarily large. This means that for a large number \(x\), the probability that a randomly chosen integer near \(x\) is prime is about \(1/\ln(x)\).

The idea was first conjectured in the late 18th century by Adrien-Marie Legendre (1798) and Carl Friedrich Gauss (1792), based on empirical evidence from tables of primes. They both proposed that \(\pi(x)\) is approximately \(x/(\ln(x) – C)\) for some constant C. However, proving this relationship required significant advances in mathematics, particularly in complex analysis. The first rigorous proofs were independently achieved by Jacques Hadamard and Charles-Jean de la Vallée Poussin in 1896. Their proofs were non-elementary, relying crucially on the properties of the Riemann zeta function in the complex plane, specifically showing it has no zeros on the line where the real part is 1.