Prime Number Theorem
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Introduction
The aim of this article is to present the proof of the prime number theorem that says that the number of primes not exceeding
is approximately
or, more precisely, that
Unfortunately, all known proofs of the prime number theorem are quite involved and require knowledge of some college level material.
We shall start with some elementary observations due to Chebyshev.
Chebyshev's estimates
The key observation is that is divisible by all primes
satisfying
. Indeed, every such prime appears in the numerator
and does not appear in the denominator
. Thus,
.