Prime Number Theorem
The prime number theorem may refer to any one of many different results estimating the prime counting function
This claim is simply that the relative error between
The prime number theorem, as proven here, is merely a consequence of a closely related result for the second Chebyshev function. As such, the main work involved in proving the theorem comes from that that theorem. Here we just show that they are equivalent.
Proof
From this result we have
For a lower bound, we have for any
noting that
Combining these and dividing by
It is then clear by the pinching theorem that the prime number theorem implies
For the reverse implication, we can derive from the above inequality
and again the result follows from the pinching theorem.
Proof
First note that we have by direct comparison that
and thus
Hence by L'Hopital's Rule and the fundamental theorem of calculus we have