Nikolski's Approach to the theorems of Beurling and Nyman regarding zeros of the Riemann ζ-function

In this thesis we present the proof of a theorem by Nikolai Nikolski. This theorem leads to a more general theorem by Nikolski regarding zero free regions of the Riemann ζ-function. This theorem is an improvement on the theorems that Nyman and Beurling proved in the nineteen fifties. Nikolski’s approach uses, in addition to step function approximations introduced by Nyman, distance functions to give more flexibility, including possible numerical experiments. The introduction discusses the Riemann Hypothesis, which always surrounds any study of the Riemann ζ-function.

The background material discussed in this thesis gives all the necessary prerequi- sites for an understanding of the proof of the main theorem. Topics include infinite products, the Gamma function, the Riemann ζ-function, Fourier series and trans- forms, the Hardy spaces, reproducing kernels, and Blaschke factors. The focus will be on the Hardy spaces of the upper and right half-planes, whose properties are deduced using the Hardy space of the unit disk via the unitary mapping of Chapter 4. The Mellin transform is also introduced and plays a vital role in the main theorem proven in chapter 6.