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, [latex]\pi(x)[/latex], 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 [latex]\pi(x)[/latex] and [latex]x/\ln(x)[/latex] is small, but rather that their ratio approaches 1 as [latex]x[/latex] becomes arbitrarily large. This means that for a large number [latex]x[/latex], the probability that a randomly chosen integer near [latex]x[/latex] is prime is about [latex]1/\ln(x)[/latex].

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 [latex]\pi(x)[/latex] is approximately [latex]x/(\ln(x) – C)[/latex] 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.