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arXiv · 2410.09558

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

Masahiro Mine
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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.

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Masahiro Mine. 2025-10-08. An upper bound for the number of smooth values of a polynomial and its applications. https://arxiv.org/abs/2410.09558

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