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Let $\pi (x)$ be the number of prime numbers less than or equal to $x$.

Euclid's proof of the infinitude of primes gives a horrible lower bound of the type $ \pi (x)>> \sqrt{\log{x}} $. Somewhat surprisingly, Euler's proof of the divergence of the sum of reciprocals of primes only gives $ \pi (x)>> \log{x}$ (see problem 8 of http://www.math.uiuc.edu/~hildebr/531.fall05/hw3.pdf).

Was anybody able to get a lower bound better than $\log{x}$ before Chebyshev got the order of magnitude right in ~ 1850?

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    Judging by http://mathworld.wolfram.com/PrimeCountingFunction.html I suspect Gauss knew this pre-1850.2012-11-10
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    Yes, Gauss knew and I find it very likely that he was able to prove a better bound than Euler's. But one can still ask whether someone did before Gauss, or whether someone *published* a proof before Chebyshev. (Gauss almost doesn't count - so much of his thinking he didn't communicate.)2012-11-10
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    @Jonah, I agree, while I also believe that Gauss knew it, one might question how he knew it. The MathWorld article says only that he postulated it.2012-11-10
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    I'm sorry I've made an incorrect statement, I could not edit it so I have removed it.2012-11-10

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