Exact divisors of polynomials with prime variable

Abstract

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.