The Prime Number Theorem describes the asymptotic distribution of prime numbers among the integers. It states that the prime-counting function [latex]\pi(x)[/latex], which gives the number of primes less than or equal to [latex]x[/latex], is asymptotically equivalent to [latex]x / \ln(x)[/latex]. Formally, [latex]\lim_{x \to \infty} \frac{\pi(x)}{x/\ln(x)} = 1[/latex]. This provides a fundamental link between primes and the natural logarithm.
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, [乳胶]\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.
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