# Month: December 2018

• ## Chebyshev’s Almost Prime Number Theorem

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 $x$ is asymptotically equal to $\frac{x}{\ln x}$.

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 $0 < c_1 < 1 < c_2$ such that, for all sufficiently large real numbers X, $c_1 \frac{X}{\log X} \leq \pi(X) \leq c_2 \frac{X}{logX}$.

Chebyshev’s Theorem

In this post, I will reproduce this proof from [Cheb] together with some comments and tables.

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