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September 2016 Exact divisors of polynomials with prime variable
Eira J. Scourfield
Funct. Approx. Comment. Math. 55(1): 83-104 (September 2016). DOI: 10.7169/facm/2016.55.1.6


In 1952 Paul Erdős obtained upper and lower bounds of the same order of magnitude for the number $N(x)$ of divisors of an irreducible polynomial $f(n)$ with integer coefficients for $n$ up to $x$; an asymptotic formula for $N(x)$ when $f$ has degree at least $3$ has not yet been established. However progress has been made in the corresponding problem when the divisors of $f(n)$ are restricted in some way and $f$ is not necessarily irreducible. In this paper we consider a~polynomial $f$ $ $with integer coefficients that may not be irreducible or squarefree. Our aim is to obtain an asymptotic formula for the number of exact divisors up to $y$ of $f(p)$ for $p$ a prime less than $x$ with $y$ as large as possible in terms of $x$. We utilize the result that Vaughan established for his elementary proof of the Bombieri-Vinogradov Theorem.


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Eira J. Scourfield. "Exact divisors of polynomials with prime variable." Funct. Approx. Comment. Math. 55 (1) 83 - 104, September 2016.


Published: September 2016
First available in Project Euclid: 19 September 2016

zbMATH: 06862554
MathSciNet: MR3549014
Digital Object Identifier: 10.7169/facm/2016.55.1.6

Primary: 11N37
Secondary: 11N32, 11N64

Rights: Copyright © 2016 Adam Mickiewicz University


Vol.55 • No. 1 • September 2016
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