The Hardy--Ramanujan inequality for sifted sets and its applications
Steve Fan
TL;DR
This work unifies sieve theory and probabilistic methods to develop a weighted Hardy–Ramanujan inequality on sifted sets. By allowing nonnegative multiplicative weights in a broad class 𝔐(A1,A2) and leveraging Pollack's sieve bound, it derives precise upper bounds for the distribution of ω(n,E) and Ω(n,E) across sifted sets, including large-deviation estimates. The authors then apply these results to a range of problems: the Erdős multiplication table, divisors of shifted primes, and the image of the Carmichael λ-function, as well as weighted normal orders for ω(s(n)). They further extend the framework to shifted primes with large divisors, obtaining new bounds and corroborating weighted instances of Erdős–Granville–Pomerance–Spiro-type conjectures, thereby enriching both the theory and its applications in multiplicative number theory.
Abstract
The well-known Hardy--Ramanujan inequality states that if $ω(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$, \[\#\{n\le x\colonω(n)=k\}\ll\frac{x(\log\log x+C)^{k-1}}{(k-1)!\log x}.\] A myriad of generalizations and variations of this inequality have been discovered. In this paper, we establish a weighted version of this inequality for sifted sets, which generalizes an earlier result of Halász and implies Timofeev's theorems on shifted primes. We then explore its applications to a variety of intriguing problems, such as large deviations of $ω$ on subsets of integers, the Erdős multiplication table problem, divisors of shifted primes, and the image of the Carmichael $λ$-function. Building on the same circle of ideas, we also generalize Troupe's result on the normal order of $ω(s(n))$ for the sum-of-proper-divisors function $s(n)$, confirming for the first time the weighted version of a special case of a 1992 conjecture by Erdős, Granville, Pomerance, and Spiro.