Year: 2026
Type: preprint
Abstract: Let $P_1(n)$ and $P_2(n)$ be the largest and second-largest distinct prime factors of $n$. Alladi and Johnson asked whether there is a bounded function $f$ on the primes such that the averages $\frac{1}{x}\sum_{2\le n\le x}f(P_1(n))$ and $\frac{1}{x}\sum_{2\le n\le x}f(P_2(n))$ converge to different limits, where $f(P_2(n))=0$ when $n$ is a prime power. We show that this cannot happen. In fact, whenever the first average converges, the second also converges to the same limit.
Source: arXiv (Cornell University)
Author: Chen, Dijia
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7201845149