Jori Merikoski

Jori Merikoski, Département de Mathématiques et de Statistiques, Université de Turku, Finlande


Large prime factors on short intervals

We show that for all large enough x the interval [x, x + x1/2 log1.39 x] contains numbers with a prime factor p > x18/19.
Our work builds on the previous results of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same
problem for the longer intervals [x, x + x1/2+ε], where the main tools used are Harman's sieve method and mean value
estimates for Dirichlet polynomials. The main new ingredient that we use is the method of Matomäki and Radziwiłł
(2016) for bounding Dirichlet polynomial mean values, which we apply to obtain 'Type II information'. This allows us
to take shorter intervals than in the above mentioned previous works.