Extreme values of the Dedekind $\Psi$ function

Patrick Solé (1) , Michel R. P. Planat (2)

1 Télécom ParisTech
2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

Let $\Psi(n):=n\prod_{p \vert n}(1+\frac{1}{p})$ 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 $N_n=2\cdots p_n$ 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.

Mots clés: Dedekind psi function; Riemann hypothesis; Robin criterion; Primorial number

Domaines: Théorie des nombres [math.NT]

Patrick Solé, Michel R. P. Planat. Extreme values of the Dedekind $\Psi$ function. Journal of Combinatorics and Number Theory, 2011, 3 (1), pp.33-38. ⟨hal-00533801v2⟩