On Robin's criterion for the Riemann Hypothesis

TL;DR

This paper provides elementary and number-theoretic insights into Robin's criterion for the Riemann Hypothesis, showing that potential counterexamples must have specific divisibility properties, thus narrowing the search for violations.

Contribution

It proves that any n<37 not satisfying Robin's inequality must be even, non-squarefree, non-squarefull, and divisible by a fifth power, refining the conditions for potential counterexamples.

Findings

Counterexamples must be divisible by a fifth power >1

n<37 not satisfying Robin's inequality are even and have specific factorizations

RH holds iff all numbers divisible by a fifth power satisfy Robin's inequality

Abstract

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d<e^{gamma}n loglog n is satisfied for n>=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.