Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis

TL;DR

This paper presents a new elementary reformulation of the Riemann Hypothesis using Robin's theorem, characterizing it through a specific inequality involving the divisor sum function and composite numbers.

Contribution

It introduces a novel criterion for the Riemann Hypothesis based on properties of G(n) and superabundant numbers, providing an elementary perspective.

Findings

Riemann Hypothesis equivalent to a unique inequality involving G(n) for composite N

Uses Robin's and Gronwall's theorems to establish the equivalence

Provides an alternative proof involving properties of superabundant numbers

Abstract

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erd\H{o}s's results.