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Prime number theorem

Number theory

The theorem that the number of primes not exceeding x is asymptotic to x / ln x, meaning the ratio of the two tends to 1 as x grows.

What is usually left out

Asymptotic equality is weaker than it looks: the ratio tending to 1 permits the absolute difference to grow without bound, and it does. The logarithmic integral li(x) is a substantially better approximation than x / ln x, and how well it does is exactly what the Riemann hypothesis pins down.

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