Large prime factors on short intervals

TL;DR

This paper proves that for sufficiently large x, short intervals of length about x^{1/2} times a logarithmic factor contain numbers with large prime factors exceeding x^{18/19}, advancing understanding of prime factors in small intervals.

Contribution

It introduces an iterative method based on Matomäki and Radziwiłł's work to analyze prime factors in shorter intervals, improving previous bounds and incorporating sieve techniques.

Findings

Intervals of length x^{1/2} log^{1.39} x contain numbers with prime factors > x^{18/19}

New iterative approach enhances bounds on prime factors in short intervals

Method avoids loss of logarithmic powers in sieve applications

Abstract

We show that for all large enough $x$ the interval $[x,x+x^{1/2}\log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals $[x,x+x^{1/2+\epsilon}].$ We also incorporate some ideas from Harman's book `Prime-detecting sieves' (2007). The main new ingredient that we use is the iterative argument of Matom\"aki and Radziwi{\l}{\l}(2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of $\log x$ when applying Harman's sieve method.