Prime Number Theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).

The first such distribution found is $π (N) ~ N / log(N)$ , where $π (N)$ is the prime-counting function and $log(N)$ is the natural logarithm of $N$ . This means that for large enough $N$ , the probability that a random integer not greater than $N$ is prime is very close to $1 / log(N)$ . Consequently, a random integer with at most $2 n$ digits (for large enough $n$ ) is about half as likely to be prime as a random integer with at most $n$ digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime ( $log(10 1000) \approx 2302.6$ ), whereas among positive integers of at most 2000 digits, about one in 4600 is prime ( $log(10 2000) \approx 4605.2$ ). In other words, the average gap between consecutive prime numbers among the first $N$ integers is roughly $log(N)$ .

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