Prime Number Theorem

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The number of prime numbers below x is asymptotic to $x/\log~x$ or, equivalently, to the logarithmic integral $Li(x).$ This fact is known as the "prime number theorem"; it was proved in the early 20th century by Hadamard and de la Vallee-Poussin.

Informally and handwavily: "the probability that n is prime is approximately $1/\log(n).$ " (Of course this statement is nonsense if taken at face value, but for many purposes the prime numbers behave rather like random numbers selected with that density.)

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