Robin’s Criterion on Divisibility (II)

Abstract

The Riemann hypothesis, renowned for its deep connection to the distribution of prime numbers, remains a central problem in mathematics. Understanding the distribution of primes is crucial for developing efficient algorithms and advancing our knowledge of number theory. The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. Several equivalent formulations of the Riemann hypothesis exist. Robin's criterion for the Riemann hypothesis is based on an inequality that divisor sum function $\sigma$ must satisfy at natural numbers greater than 5040. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. By using Robin's criterion on superabundant numbers, we present a novel approach that culminates in a complete proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.