On the greatest prime factor and uniform equidistribution of quadratic polynomials

Abstract

We show that the greatest prime factor of $n^2+h$ is at least $n^{1.312}$ infinitely often. This gives an unconditional proof for the range previously known under the Selberg eigenvalue conjecture. Furthermore, we get uniformity in $h \leq n^{1+o(1)}$ under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. We also prove a variant of the divisor problem for $ax^2+by^3$, which was used by the second author to give a conditional result about primes of that shape.