An upper bound for the number of smooth values of a polynomial and its applications

TL;DR

This paper establishes a new upper bound on the count of smooth values taken by integer polynomials, improving previous results and enabling applications in number theory, including zeros of the Hurwitz zeta-function and primitive divisors of quadratic polynomials.

Contribution

It introduces a sharper upper bound for smooth polynomial values, extending prior work and providing new proofs and applications in analytic number theory.

Findings

New upper bound for smooth polynomial values

Alternative proof for zeros of Hurwitz zeta-function

Applications to primitive divisors of quadratic polynomials

Abstract

We prove a new upper bound for the number of smooth values of a polynomial with integer coefficients. This improves Timofeev's previous result unless the polynomial is a product of linear polynomials with integer coefficients. As an application, we provide another proof for a result of Cassels which was used to prove that the Hurwitz zeta-function with algebraic irrational parameter has infinitely many zeros on the domain of convergence. We also apply the main result to a problem on primitive divisors of quadratic polynomials.