Murty–Murty conjecture on the second moment M2(x)

Establish the asymptotic M2(x) ∼ C2·log x as x → ∞, where M2(x) = (1/x)·Σ_{n ≤ x} ω*(n)^2 and C2 > 0 is a constant.

Background

Moments M_k(x) of the shifted-prime divisor function encode the distribution of ω(n). While M1(x) is well understood and bounds for M2(x) are known, an exact asymptotic for M2(x) remains central to understanding the variance and higher-order behavior of ω(n).

Murty and Murty provided upper and lower bounds for M2(x), motivating the conjecture of an exact linear-in-log x asymptotic with an explicit constant C2.