S-parts of values of univariate polynomials

Abstract

Let S={p1,…,ps} be a finite non-empty set of distinct prime numbers, let f∈Z[X] be a polynomial of degree n≥1, and let S′⊆S be the subset of all p∈S such that f has a root in Zp. For any non-zero integer y, write y=p1k1…psksy0, where k1,…,ks are non-negative integers and y0 is an integer coprime to p1,…,ps. We define the f-normalized S-part of y by [y]f,S:=p1k1rp1,S(f)…psksrps,S(f), with rp,S(f)=1 if p∈S∖S′ and rp,S(f)=RS′(f)/Rp(f) if p∈S′, where Rp(f) denotes the largest multiplicity of a root of f in Zp and RS′(f):=maxp∈S′⁡Rp(f). For positive real numbers ε,B with ε<RS′(f)/n, we consider the number N˜(f,S,ε,B) of integers x such that |x|≤B and 0<|f(x)|ε≤[f(x)]f,S. We prove that if s′:=#S′≥1, then N˜(f,S,ε,B)≍f,S,εB1−(nε)/RS′(f)(log⁡B)s′−1 as B→∞. Moreover, if f has no multiple roots in Zp for any p∈S′ and s′:=#S′≥2, then there exists a constant C(f,S,ε)>0 such that N˜(f,S,ε,B)∼C(f,S,ε)B1−nε(log⁡B)s′−1 as B→∞.