Eisenstein Series and Powerfree Integers

Abstract

The purpose of this thesis is to use techniques from analytic number theory to study two main topics: Eisenstein series and powerfree integers. First, we study the sign changes of Fourier coefficients of Eisenstein series newforms, obtaining an asymptotic for when on average, the first negative coefficient occurs. Next, we develop a general framework to determine when a set of integers A contains infinitely many k-th powerfree numbers. This approach is based on obtaining certain moment bounds for the discrete Fourier transform associated with A. Finally, in joint work with Gunther Cornelissen and Berend Ringeling, we study the zeros of Eisenstein series for Γ(N) in the standard fundamental domain of Γ(1). In particular, we show that as the weight goes to infinity, all the zeros are transcendental (except possibly at i and rho), and they converge in Hausdorff distance to a finite configuration of geodesic segments. Moreover, for odd N, we can describe the exact 'convergence speed' of the zeros to the unit circle, as well as angular equidistribution of the zeros as the weight tends to infinity.