Moments of the shifted prime divisor function

TL;DR

This paper investigates the moments of a shifted prime divisor function, confirming a conjecture by Fan and Pomerance through combinatorial and analytic number theory techniques.

Contribution

It establishes the asymptotic behavior of the moments of the shifted prime divisor function, confirming a recent conjecture.

Findings

For each integer k ≥ 2, the sum of ω*(n)^k up to x is asymptotically proportional to x(log x)^{2^k - k - 1}.

The proof introduces a combinatorial identity related to least common multiples as a multiplicative inclusion-exclusion analogue.

The results deepen understanding of the distribution of shifted prime divisors and their moments.

Abstract

Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent conjecture of Fan and Pomerance. Our proof uses a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, along with analytic tools from number theory.