Recent zbMATH articles in MSC 11N25https://www.zbmath.org/atom/cc/11N252021-02-27T13:50:00+00:00WerkzeugThe distribution of divisors of polynomials.https://www.zbmath.org/1453.111222021-02-27T13:50:00+00:00"Ford, Kevin"https://www.zbmath.org/authors/?q=ai:ford.kevin-b"Qian, Guoyou"https://www.zbmath.org/authors/?q=ai:qian.guoyouSummary: Let \(F(x)\) be an irreducible polynomial with integer coefficients and degree at least 2. For \(x\ge z\ge y\ge 2\), denote by \(H_F(x,y,z)\) the number of integers \(n\le x\) such that \(F(n)\) has at least one divisor \(d\) with \(y<d\le z\). We determine the order of magnitude of \(H_F(x,y,z)\) uniformly for \(y+y/\log^Cy<z\le y^2\) and \(y\le x^{1-\delta}\), showing that the order is the same as the order of \(H(x,y,z)\), the number of positive integers \(n\le x\) with a divisor in \((y,z]\). Here \(C\) is an arbitrarily large constant and \(\delta>0\) is arbitrarily small.