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 !