We know that the average of $\phi(n)/n$ is approximated by a constant. Here $\phi $ is the Euler quotient function. One can furthermore show asymptotics with a secondary main term, at least for the smooth sum $$ \sum_{n \in \mathbb{N} } \frac{\phi(n) } {n} w(n/x)=c_0(w) x+c_1(w) (\log x ) +o(\log x ) ,$$ where $w$ is a smooth weight and $c_0,c_1$ are constants depending on $w$.

Can we prove asymptotics for the secondary term regarding the shifted sum $$ \sum_{n \in \mathbb{N} } \frac{\phi(n) } {n}\frac{\phi(n+1) } {n+1} w(n/x) $$ for some $w$? It is not clear to me whether the secondary term here should be oscillating or like $\sim c \log^2 x$ or something else. The standard approach to prove the previous asymptotic relies on the fact that $\frac{\phi(n) } {n} $ is multiplicative, whereas $\frac{\phi(n) } {n}\frac{\phi(n+1) } {n+1}$ is clearly not.