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

TL;DR

The paper establishes a new, explicit upper bound for the number of $y$-smooth values produced by an integer-coefficient polynomial, refining Timofeev's bound except when the polynomial is a product of linear factors. The core technique is an inductive, parameterized control of the difference $\Psi_f(x,y)-\Psi_f(z,y)$ using decompositions into divisor sums and an arithmetic function $\\omega_f$, with carefully chosen auxiliary quantities $V$ and $W$. The bound features an explicit function $\\gamma_f(u)$, and in particular ensures $\\gamma_f(u)<1$ when $f$ has a non-linear factor, yielding concrete improvements and even a universal bound $\\gamma_f(u)\le 0.913...$ for irreducible degree $d\ge2$. The results have notable applications: a new proof of Cassels’ connection between smooth values and zeros of the Hurwitz zeta-function with algebraic irrational parameter, a lower bound for $C_\\alpha(x)$ for algebraic irrationals, and a stronger lower bound for the primitive-divisor count $R_b(x)$ of $n^2+b$, with additional links to arctan irreducibility and related arithmetical questions.

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.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Proposition 2.1
• proof
• Lemma 3.1
• proof