Extreme values of the Dedekind \(\Psi\) function (Q2905233)

Define the Dedekind \(\Psi\) function by NEWLINE\[NEWLINE\Psi(n)= n\prod_{p|n} \Biggl(1+{1\over p}\Biggr)NEWLINE\]NEWLINE and let \(R(n)= {\Psi(n)\over n\log\log n}\). The authors show that \(R(n)< e^\gamma\) for all \(n\geq 31\). Let \(N_n= \prod^n_{k=1} p_k\) be the primorial number of order \(n\), where \(p_k\) is the \(k\)th prime. It is established unconditionally that \(R(N_n)> {e^\gamma\over\zeta(2)}\) for infinitely many \(n\), and that this inequality holds for all \(n\geq 3\) if and only if the Riemann hypothesis is true. The proofs depend on results of \textit{J.-L. Nicolas} [J. Number Theory 17, 375--388 (1983; Zbl 0521.10039)].