On a Series Involving Euler's Function

Abstract

The goal of this thesis is to provide an in-depth analysis and discussion of an equivalence to the Riemann Hypothesis (RH) proven by Jean-Louis Nicolas. Nicolas’ proof relates RH to an inequality of Euler’s totient function φ, and establishes a number-theoretic equivalence to RH. If Nicolas’ criterion holds for all primorial numbers, then RH is true. If not, then RH is false. This proof is given an original translation into English from French and annotated, with small corrections to computations and commentary when deemed necessary. His work is then extended by relating the equivalence to the convergence of an infinite series which is shown to converge to 1/2. Using this series and the related partial sum, consequences of the truth or falsehood of RH are explored in the context of Nicolas’ criterion. We assume both the truth and falsehood of RH, and in doing so underscore the extreme difficulty of this problem as well as the delicacy of the inequalities involved. Also provided are multiple programs which computationally verify expectations regarding different quantities from the analytic results section. Optimization of these programs are discussed as well as difficulties. These programs produce plots of the behavior of consequential arithmetic-valued functions, which are included in Chapter 4. The research results were limited by the nature of the problem. None of the analysis on the convergence criterion yielded a contradiction to an established result or conjecture, assuming either RH true or false. However, RH is known to be one of the most difficult problems in modern mathematics and significant progress was largely outside the scope of this thesis. The hope is that this research renews interest into Nicolas’ criterion specifically and arithmetic inequalities equivalent to RH in general.