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PreprintarXiv licencearXiv

Rigidity of Averages over the Two Largest Prime Factors

  • Dijia Chen
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Abstract

Let P1(n) and P2(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 1/x∑2≤ n≤ xf(P1(n)) and 1/x∑2≤ n≤ xf(P2(n)) converge to different limits, where f(P2(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.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026rigidity,
  title = {{Rigidity of Averages over the Two Largest Prime Factors}},
  author = {Dijia Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.05191},
  eprint = {2608.05191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.05191},
}
RIS
TY  - UNPB
TI  - Rigidity of Averages over the Two Largest Prime Factors
AU  - Dijia Chen
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2608.05191
UR  - https://arxiv.org/abs/2608.05191
AB  - 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 1/x∑_2≤ n≤ xf(P_1(n)) and 1/x∑_2≤ n≤ xf(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.
ER  -