Vendredi 14 janvier 2011 à 15h00 en salle C47
Patrick Solé (Télécom ParisTech)

Titre : $\sigma,\varphi,\psi$ and RH

Résumé :
In 1983 Jean Louis Nicolas proved that the inequality $$\frac{N_k}{\varphi(N_k)} >e^\gamma \log \log N_k,$$ where \begin{itemize} \item $\gamma \approx 0.577$ is the Euler Mascheroni constant, \item $\varphi$ Euler totient function ,\item $N_n=\prod_{k=1}^np_k$ the primorial of order $n,$ \end{itemize} holds for all $k \ge 1$ if RH is true \cite[Th. 2 (a)]{N}. Conversely, if RH is false, the inequality holds for infinitely many $k,$ and is violated for infinitely many $k$ \cite[Th. 2(b)]{N}. We give an analogue for Dedekind $\psi$ function defined by $$\Psi(n):=n\prod_{p \vert n}(1+\frac{1}{p}),$$ and more generally for the function $$\psi_b (n)=n\prod_{p \vert n}\frac{1-\frac{1}{p^b}}{1-\frac{1}{p}},$$ for $b$ real $>1,$ that occurs in the KMS states of Bost Connes quantum model of prime numbers \cite{Connes1995}. The special case of $b$ integer allows us to bound the sum of divisors function $\sigma$ for $b-$free integers. Thus we can prove that Robin inequality \cite{R} given by $$\sigma(n) =\sum_{d\vert n} d< e^\gamma n \log \log n,$$ holds for $7$-free integers $n.$ This is joint work with Michel Planat.

Remarques : Attention à l'heure !