On the largest prime factor of integers in short intervals III

TL;DR

This paper proves that for large x, short intervals contain integers with very large prime factors, improving previous bounds and solving a specific exercise in Harman's monograph.

Contribution

It introduces a new method combining Watt's mean value theorem and sieve techniques to improve bounds on prime factors in short intervals.

Findings

Interval [x, x + x^{1/2 + ε}] contains integers with prime factors > x^{35/36 - ε}

Improves previous bound from 1/26.5 to 1/36

Provides a solution to a problem in Harman's monograph

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 $\gamma = \frac{1}{36}$ to the Exercise 5.1 in Harman's monograph and improves the previous record of the author proved in 2024, where $\gamma = \frac{1}{26.5}$ is obtained.