Title:Large prime factors on short intervals

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 Matomäki and Radziwiłł(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.