Résumé : T.C. Brown and A.R. Freedman proved that the set ${\mathcal{P}}_2$ of products of two primes contains no dense cluster; technically, ${\mathcal{P}}_2$ has a zero upper Banach density, defined as ${\delta }^{\ast }({\mathcal{P}}_2)=\underset{H\mapsto \mathrm{\infty }}{lim}\underset{x\mapsto \mathrm{\infty }}{lim sup}\frac{1}{H}Card\{n\in {\mathcal{P}}_2:x<n\le x+H\}$.
Pramod Eyyunni, Sanoli Gun and I jointly studied the local behaviour of the product of two shifted primes ${\mathcal{Q}}_2=\{(q-1)(r-1):q,rprimes\}$. Assuming a classical conjecture of Dickson, we proved that ${\delta }^{\ast }({\mathcal{Q}}_2)=1/6$. Notice that we know no un-conditional proof that ${\delta }^{\ast }({\mathcal{Q}}_2)$ is positive. The application, which was indeed our motivation, concerns the study of the local behaviour of the set $\mathcal{V}$ of values of Euler's totient function. Assuming Dickson's conjecture, we prove that ${\delta }^{\ast }(\mathcal{V})\ge 1/4$. The converse inequality ${\delta }^{\ast }(\mathcal{V})\le 1/4$ had been proved in the previous millenium by K. Ford, S. Konyagin and C. Pomerance.

Keywords : Banach density; products of shifted primes; values of Euler's function; Dickson's conjecture