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Let $\pi(x)$ be the prime counting function. It is well-known that $$\lim_{x\to\infty}\frac{\pi(x)}{x}=0.$$ My question is who is the first person to prove this result? Did Eratosthenes prove this theorem for the first time?

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    $\begingroup$ This was not known until Chebyshev proved a stronger result (1849). $\endgroup$ Commented Jan 7 at 14:37
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    $\begingroup$ Ancient Greeks did not entertain propositions of this sort, even as a conjecture (of Legendre and Gauss), it only appeared at the very end of 18th century. Wikipedia has a detailed History of the proof of the asymptotic law of prime numbers. $\endgroup$
    – Conifold
    Commented Jan 8 at 3:35

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