In their article "On the spectral radius of a {0,1) Matrix Related to Mertens' Function", Barret et al. assert an inequality just at the end of p. 156. Apparently, this inequality comes from the Abel summation formula applied to
$C(n, k+1) = \sum_w ( C(w, k) - C(w, k-1)) ([n/w] - 1)$,
which yields in this case to
$C(n,k+1) = \sum_w C(w,k) ([n/w] - [n/(w+1)])$.
And apparently, they allow themselves to replace the term $[n/w]-[n/(w+1)]$ by $n/w - n/(w+1)$, upon transforming the equality into an inequality. That is, they consider that
$[n/w]-[n/(w+1)] \leq n/w - n/(w+1)$.
But this is false, as can be seen for example with n = 10 and w = 3 or n = 15 and w = 3. Am I missing something?
The key here is to perform the two steps (Abel transform, and dealing with the floors) in the other order.
Namely, first, handle the floors: $$ \sum_{w} c(w,k) \left( \left\lfloor \frac{n}{w} \right\rfloor - 1\right) \leq \sum_{w} c(w,k) \cdot \frac{n}{w} $$ and then perform the Abel transform. This way, you avoid the troublesome (and, indeed, false) inequality at the end.