On the sum of least prime factors in short intervals

@misc{zhang2026least,
  title = {{On the sum of least prime factors in short intervals}},
  author = {Xu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.24930},
  eprint = {2608.24930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.24930},
}

TY  - UNPB
TI  - On the sum of least prime factors in short intervals
AU  - Xu Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2608.24930
UR  - https://arxiv.org/abs/2608.24930
AB  - Let p(n) denote the least prime factor of n and L_C(x)=Cx^1/2(log x)^2 with C>0. The sum of p(n)/n over composite n lying in the short interval [x, x+L_C(x)], a question raised by Erdős and Graham, is studied. (i) The constant c=8 in the mean asymptotic is estimated S(x)=∑_n<x, n compositep(n)/n=(c x^1/2)/((log x)^2)(1+O(1/(log x))) +O(x^1/3/(log x)). (ii) For every fixed C>0, the window sums μ_C(x):=∑_x≤ n≤ x+L_C(x)p(n)/n over composites have mean 4C: 1/X∑_x≤ Xμ_C(x)=4C+O_C(1/log X), and second moment 1/X∑_x≤ X(μ_C(x)−4C)^2=O_C(1). (iii)Under the Maier-consistent bounded-fluctuation hypothesis, μ_C(x)=4C+O_C(1) holds uniformly in x, and the Erdős--Graham question has an affirmative answer. Unconditionally, the uniform statement remains open; proving it would require resolving short-interval prime estimates at scale (log y)^2.
ER  -