On the largest prime factor of integers in short intervals III

Using Watt’s mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x,x+x^{\frac{1}{2}+\epsilon }]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\epsilon }$ 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.