Linnik's Proof of the Waring-Hilbert Theorem

Abstract: In number theory, Waring–Hilbert’s theorem guarantees that for each k there is an integer h ≥ 0 so that, for every non-negative integer n there are non-negative integers a1,a2,...,ak such that ak 1 + ak 2 + ··· + ak h = n. In this thesis the problem will first be proved in the specific case where k = 2. Then the proof of the general case due to Yuri Linnik will be given. The notion of Shnirelman density will be introduced. Although the approach for proving Waring–Hilbert’s theorem is elementary, several methods from various fields of mathematics will be used. Instances of the Riemann zeta function will be used in order to show that h is finite. The lasts steps in the argument will be carried out with the aid of Fourier series.