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.[-]

The Liouville function $\lambda(n)$ takes the value +1 or -1 depending on whether $n$ has an even or an odd number of prime factors. The Liouville function is closely related to the characteristic function of the primes and is believed to behave more-or-less randomly.
I will discuss my very recent work with Radziwill, Tao, Teräväinen, and Ziegler, where we show that, in almost all intervals of length $X^{\varepsilon}$, the Liouville function does not correlate with polynomial phases or more generally with nilsequences.
I will also discuss applications to superpolynomial number of sign patterns for the Liouville sequence and to a new averaged version of Chowla's conjecture.[-]