On the greatest prime factor and uniform equidistribution of quadratic polynomials

2025/05/01 by Grimmelt, Lasse, Merikoski, Jori · 1 citation
#11N32 #11N75 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.00493

Abstract

We show that the greatest prime factor of n²+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 ≤ 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 ax²+by³, which was used by the second author to give a conditional result about primes of that shape.

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