Chebychev's Theorem states that the order of magnitude of the prime counting function $\pi(x)$ is $x/\log x$. This result was "the state of the art" until the proof of the Prime Number Theorem by Hadamard and de la Vallée-Poussin in 1895. We shall outline a proof of Chebychev's estimates and deduce from his estimate Mertens' Theorem which provides an asymptotic formula for $\sum_{p\le x}p^{-1}$ as $x\rightarrow\infty$.
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[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/4234
[3] https://www.mpim-bonn.mpg.de/node/7671