Preventing Exceptions to Robins InEquality

Abstract

For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin's InEquality $X(n):=\frac{\sigma (n)}{n\ln \ln n}<e^{\gamma }$ (RIE). The largest known violation of RIE is $n_8=5040$. In this paper Robin's multipliers are split into logarithmic terms $\mathcal{L}$ and relative divisor sums $\mathcal{G}$. A violation of RIE above $n_8$ is proposed to imply oscillations that cause $\mathcal{G}$ to exceed $\mathcal{L}$. To this aim Alaoglu and Erdős's conjecture for the CA numbers algorithm is used and the paper's key points are in section 4.2

22 pages (12 without tables and directories), 6 figures, 2 tables. V2 addresses some typographical bugs, V3 added outline

References in corpus (4)

• Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method
• On Robin's criterion for the Riemann Hypothesis
• The Nicolas and Robin inequalities with sums of two squares
• Proof of the strong Density Hypothesis