On the largest prime factor of integers in short intervals III
Runbo Li
TL;DR
The paper proves that for sufficiently large $x$, the interval $[x, x+x^{1/2+\varepsilon}]$ contains an integer with a prime factor exceeding $x^{35/36-\varepsilon}$, attaining γ = 1/36 in Harman's Exercise 5.1. It achieves this by combining Watt's mean-value theorem with a delicate sieve decomposition, implemented via Dirichlet polynomials $M(s)$, $N(s)$, $H(s)$, and $L(s)$ and a multi-layer Buchstab analysis. A central innovation is a refined Type I/II information bundle and a series of Buchstab-identity-based decompositions, including a novel reversed Buchstab step to manage almost-prime components, all organized around a precise region- and loss-tracking framework. The result improves the previous record and advances understanding of large-prime-factor structure in short intervals, with implications for approximating primes through large factors and for related Harman-type problems. All along, the analysis maintains strict control of error losses, culminating in an explicit lower bound that guarantees positivity of the main sieve term.
Abstract
Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently large $x$. This gives a solution with $γ= \frac{1}{36}$ to the Exercise 5.1 in Harman's monograph and improves the previous record of the author proved in 2024, where $γ= \frac{1}{26.5}$ is obtained.