Belirli bir büyüklükten küçük asal sayılar üzerine

In this study, we give different proofs of the Prime Number Theorem which gives an estimate on the number of primes not exceeding x, for a given real number x. We first prove the theorem with elementary methods in which we do not get use of any complex function theory. This proof does not give an error term but rather gives an asymptotic to the function counting primes up to x. In the second part we give two analytic proofs of the PNT with exponential error terms, the last one providing a better error term. Many of the properties of the Riemann zeta-function are studied since we use them frequently along the way. Finally we give the PNT for arithmetic progressions which gives an estimate on the number of primes not exceeding x belonging to a certain arithmetic progression. The role of the Dirichlet L-functions serves as an analogue to Riemann zeta-function's role in the proof of PNT. So we study the properties of the Dirichlet L-functions as well.