New results for witnesses of Robin’s criterion

Abstract

In a seminal paper, Robin proved that the Riemann hypothesis holds if and only if the inequality σ ( n ) = ∑ d | n d > e γ n log ⁡ log ⁡ n \sigma (n)=\sum _{d|n} d > e^\gamma n \log \log n holds for all integers greater than 5040, where γ \gamma is the Euler-Mascheroni constant. In this article, we prove new results on putative violations of Robin’s criterion.