Set of Prime Numbers

Ovaj diplomski rad je fokusiran na dvije temeljne činjenice vezane za proste brojeve. Prva je da prostih brojeva ima beskonačno, a druga da im se gustoća smanjuje. Prva činjenica je, naravno, poznata još od vremena Euklida. Medutim, postoji jako puno dokaza ove tvrdnje koji nemaju sličnosti s izvornim Euklidovim dokazom. U ovom radu dajemo neke od tih dokaza: neki su direktni, neki uključuju analizu, a neki potječu iz sasvim drugih područja matematike. Točan način kako se gustoća prostih brojeva smanjuje formalno je zapisan u Teoremu o prostim brojevima. U radu su prezentirani neki rezultati i ideje koje su dovele do dokaza Teorema o prostim brojevima te je dana veza distribucije prostih brojeva s čuvenom Riemannovom slutnjom.In this master’s thesis, we are focused on two basic facts concerning prime numbers. The first fact is that there are infinitely many primes, and the second is that the density of primes thins out. The first fact has been, of course, known since at least the time of Euclid. However, there are a great many proofs of this result not related to Euclid’s original proof. In this thesis, we present some of these proofs: some are direct, some involve analysis, and some come from quite different areas of mathematics. The exact nature of how the density of primes thins out is formalized in the Prime Number Theorem. In this thesis, we present some results and ideas that have led to the proof of the Prime Number Theorem and also the relationship between the distribution of prime numbers and the famous Riemann’s conjecture