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General moments conjecture for M_k(x)

Establish, for each fixed integer k ≥ 2, the asymptotic M_k(x) ∼ C_k·(log x)^{2^k − k − 1} as x → ∞, where M_k(x) = (1/x)·Σ_{n ≤ x} ω*(n)^k and C_k > 0 depends on k.

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Background

Beyond the second moment, understanding higher moments M_k(x) is crucial to fully characterizing the distribution of ω*(n). Recent work has established the correct order of magnitude for all fixed k ≥ 2, but exact asymptotics with constants are conjectural.

The conjecture provides a precise exponent for the growth in terms of log x and posits existence of constants C_k for each k, paralleling moment asymptotics in probabilistic number theory.

References

More recently, Pomerance and the first author established the estimate $M_3(x)\asymp(\log x)4$ and made the more general conjecture that for each $k\ge2$ there exists a constant $C_k>0$ such that $M_k(x)\sim C_k(\log x){2k-k-1}$.

— The maximal order of the shifted-prime divisor function (2510.14167 - Fan et al., 15 Oct 2025) in Section 1 (Introduction)