Techniques from harmonic analysis play a crucial role in understanding problems in analytic number theory. For example, in 1916 Hermann Weyl initiated the study of equidistribution of sequences on the additive circle R/Z, establishing a connection between Fourier analysis, number theory, and dynamical systems. A quantitative study of equidistribution was later carried out by van der Corput, Pisot and others. In particular, discrepancy theory provided a means of comparing how well sequences are equidistributed. Two primary upper bounds on the discrepancy have been produced due to LeVeque and Erdos-Tur an, both Fourier analytic in nature.
Fourier analytic techniques can be extended to other locally compact abelian groups, leading to interesting number theory. In this dissertation, we look at the compact abelian group Zp of p-adic integers. We fi rst review Fourier analysis on Zp, and then use it to develop a quantitative study of equidistribution. In particular, we prove a LeVeque-type Fourier analytic upper bound on the discrepancy of sequences. We establish p-adic analogues of the classical Dirichlet and Fejer kernels on R/Z, and investigate their properties. We also prove a p-adic Fourier analytic Koksma inequality.