Praštevilski izrek

Članek obravnava dokaz praštevilskega izreka. V ta namen je predstavljena osnovna teorija neskončnih produktov in Riemannova funkcija zeta. Izpeljana je Eulerjeva produktna formula, njena meromorfna razširitev na desno polovico kompleksne ravnine in predpis za njen logaritmični odvod. Definirani sta Mangoldtova in psi funkcija. Z njuno pomočjo je poiskana ekvivalentna oblika praštevilskega izreka, ki je nazadnje dokazana z metodami kompleksne analize.

Prime number theorem

In this paper, prime number theorem is proved using analytical methods. For this purpose some theory of infinite products is introduced and the Riemann zeta function is used. The Euler product formula, its meromorphic extension on the right half of the complex plane is derived and the definition of its logarithmic derivative is found. The Mangoldt and psi functions are defined and used to find the equivalent formulation of the prime number theorem. Finally, the prime number theorem is proved using complex analysis methods.