# Primjena L-funkcija u teoriji brojeva

Patljak, Marija (2016) Primjena L-funkcija u teoriji brojeva. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

This thesis proves Dirichlet’s theorem, a result that states that any arithmetic progression of the form $kn + h \quad (h, k)=1, \quad n=0,1,2, \dots,$ contains infinitely many primes. In the first chapter we give basic definitions and properties of functions that are used in this thesis. The most important result derived in this section is the proof by Harold N. Shapiro that shows that the series $\Sigma \, p^{-1} \log p$, extended over all primes, diverges. Also, the basic theorems that show that the set of prime numbers is infinite are proven. In the second chapter we become familiar with characters of finite Abelian groups. Of great importance for the proof of Dirichlet’s theorem are Dirichlet characters $\chi$ and the functions $L(1, \chi)$. It is particularly important to know that $L(1, \chi) \neq 0$ for all real nonprincipal characters $\chi$. The proof of Dirichlet’s theorem is carried out in the third, and the last, chapter. Through series of lemmas we prove Dirichlet’s theorem as a consequence of Shapiro’s formula (from the first chapter), extended over all primes $p$ which are congruent to $h$ mod $k$.