The conjectured constant for the second moment of the shifted prime-divisor function

Define

$$M_2(x)=\frac{1}{x}\sum_{n\le x}\omega^*(n)^2\quad\text{and}\quad S_2(x)=\frac{1}{x}\#\{(p,q):[p-1,q-1]\le x\}.$$

Partial summation shows that $S_2(x)\sim C$ implies $M_2(x)\sim C\log x$.

Second-moment constant conjecture. One has

$$S_2(x)\sim\frac{105}{4\pi^2}\zeta(3),$$

and

$$M_2(x)\sim\frac{105}{4\pi^2}\zeta(3)\log x.$$

This gives the predicted constant $C=\frac{105}{4\pi^2}\zeta(3)\approx3.19709$, differing from the earlier heuristic value based on the Elliott–Halberstam conjecture. The paper notes numerical support, but the asymptotics remain unproved.