We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively.[-]

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 $\delta^{*}(\mathcal{P}_{2}) =\lim_{H\mapsto \infty} \limsup_{x\mapsto \infty} \frac{1}{H} Card \{n\in \mathcal{P}_{2}:x< n\leq x+H\}$.
Pramod Eyyunni, Sanoli Gun and I jointly studied the local behaviour of the product of two shifted primes $\mathcal{Q}_{2}=\{(q-1)(r-1):q,r \, primes\}$. Assuming a classical conjecture of Dickson, we proved that $\delta^{*}(\mathcal{Q}_{2}) = 1/6$. Notice that we know no un-conditional proof that $\delta^{*}(\mathcal{Q}_{2})$ 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 $\delta^{*}(\mathcal{V})\geq 1/4$. The converse inequality $\delta^{*}(\mathcal{V})\leq 1/4$ had been proved in the previous millenium by K. Ford, S. Konyagin and C. Pomerance.[-]