Primes and Arithmetic Progressions

Abstract

In this thesis, various result concerning arithmetic progressions and primes are proven. In the first part, I prove that every subset A of {1, ... , N} that is large enough contains an arithmetic progression of length three. In the second part, I prove an asymptotic for the number of arithmetic progressions in the primes smaller than N. The proof of the first part uses finite Fourier analysis and is elementary and self-contained. The proof of the second part is an application of the Hardy-Littlewood Circle method and uses a deep theorem concerning the distribution of primes in arithmetic progressions (the theorem of Siegel and Walfisz).