Extreme values of the Dedekind $\Psi$ function

Abstract

Let $\Psi (n):=n{\prod }_{p|n}(1+\frac{1}{p})$ denote the Dedekind $\Psi$ function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{\Psi (n)}{n\log \log n}.$ We prove unconditionally that $R(n)<e^{\gamma }$ for $n\ge 31.$ Let $N_n=\mathrm{2...}{p}_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\frac{e^{\gamma }}{\zeta (2)}$ for $n\ge 3$ is equivalent to the Riemann Hypothesis.

5 pages, to appear in Journal of Combinatorics and Number theory