Extreme values of the Dedekind $\Psi$ function

TL;DR

This paper investigates the extremal behavior of the Dedekind ta function, establishing bounds and linking its properties to the Riemann Hypothesis through the analysis of specific ratios and primorials.

Contribution

It proves an unconditional upper bound for the ratio R(n) and shows the equivalence between a particular inequality involving primorials and the Riemann Hypothesis.

Findings

Unconditionally, R(n) < e^3 for n  31.

The inequality R(N_n) > e^3 / 62 for n  3 is equivalent to the Riemann Hypothesis.

Establishes a new criterion linking Dedekind ta function extremal values to the Riemann Hypothesis.

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 $N_n=2...p_n$ 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.