Fourier analysis on groups and Dirichlet' s theorem

In the first chapter we will develop the theory of Fourier analysis in locally compact Abelian groups. The theory in this chapter will be an extension of the classical of the Fourier transform, hence we will encounter some well-known properties in addiction to the Inversion theorem and the Plancherel theorem.In the second chapter we will prove the Dirichlet theorem, named after Peter Gustav Lejeune Dirichlet. The theorem states that for any k and q relatively prime positive integers, there are infinitely many primes of the form q + nk, where n is also a positive integer. The theorem represents the beginning of rigorous analytic number theory.In the third chapter we will show an elementary proof of Dirichlet¿s theorem provided by Atle Selberg in 1949. The proof is ¿elementary¿ in the sense that no complex analytic techniques are used.