Moments of the shifted prime divisor function

Abstract: Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$ denote the ``shifted prime divisor'' function. It is easy to see that $\sum_{n\leq x}\omega^*(n)=x\log\log x+O(x)$, similar to the average value of $\omega(n)$, the number of prime divisors of $n$. We confirm a recent conjecture of Fan and Pomerance by proving that, for each integer $k\geq2$, $ \qquad \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $ where the implied constant may depend only on $k$. The proof relies on a combinatorial identity for the least common multiple, viewed as a multiplicative analogue of the inclusion-exclusion principle, together with the theory of multiplicative functions.

number theory

Audience: researchers in the topic

( paper )

Combinatorial and additive number theory seminar (CANT 2026)