Powerfree Values of Polynomials

TL;DR

This paper establishes an asymptotic formula for counting k-th power free values of certain irreducible polynomials, improving previous bounds and extending results to prime arguments.

Contribution

It improves the range of k for which the asymptotic formula holds and extends the analysis to polynomial values at prime arguments.

Findings

Asymptotic formula for k-th power free values of f(n) for n up to x

Extension of results to polynomial values at prime arguments

Improved bounds on k relative to polynomial degree d

Abstract

For irreducible integer polynomials $f(n)=n^d+c$ we prove an asymptotic formula for the number of $k$-th power free values taken by $f(n)$, for $n$ running up to $x$, subject to the condition $k\ge (5d+3)/9$. This improves earlier results in which the condition was $k\ge (3d+1)/4$. We also show that one can handle $f(p)$ for prime arguments $p$, for the same range of $k$.