On the Lagarias Inequality and Superabundant Numbers

TL;DR

This paper investigates the Lagarias inequality related to the Riemann Hypothesis, demonstrating that potential counterexamples must be superabundant numbers, thus narrowing the search for violations.

Contribution

It introduces a continuous extension of harmonic numbers and proves the monotonicity of a related sequence, linking counterexamples to superabundant numbers.

Findings

The sequence B_n is strictly increasing for n ≥ 1.

Counterexamples to the Lagarias inequality, if any, are superabundant numbers.

Verification of the inequality on superabundant numbers suffices to test the hypothesis.

Abstract

We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for $n\ge 1$. As a consequence, if the Lagarias inequality has counterexamples, then the least counterexample must be a superabundant number; equivalently, it suffices to verify the inequality on the superabundant numbers.