To the right, you can see a picture of the Prime Number Theorem. It states that the number of primes up to a real number 
And this was Pafnuty Lvovich Chebyshev who almost managed to prove it around the year 1850. His almost-proof resulted in a theorem named after him.
I was recently trying to understand the proof of Chebyshev’s theorem:
Theorem 1. There are constants
Chebyshev’s Theoremsuch that, for all sufficiently large real numbers X,
.
In this post, I will reproduce this proof from [Cheb] together with some comments and tables.
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