Prime Number Theorem

The prime number theorem is broadly considered one of the greatest theorems proved in all of number theory, taking more than a hundred years between being conjectured and being proved. This thesis examines the analytic proof of the theorem. An auxiliary function to Tchebychev's ψ function is used in conjunction with a theorem showing the asymptotic behavior of the function is equivalent to the asymptotic behavior in the prime number theorem. With these tools, it is then sufficient to show that ψ1 is asymptotic to x^(2)/2. Accomplishing this is surprisingly involved and requires producing an integral that connects the ψ function to the ζ function, integrating along the real line in the complex plane whose real part is greater than 1, acquiring estimates of the ζ function and its derivative, and finally ensuring ζ has no zeroes on the real line in order to shift the contour to the real line s=1