A distribution problem for powerfree values of irreducible polynomials (Q5948340)

Let \(f(x)\) be an irreducible polynomial of degree \(d\), defined over the integers. Suppose \(f\) has no fixed \(k\)-th power factor, and let \(s_1 < s_2 < \dots\) be the sequence of values \(f(n)=s\) which are \(k\)-th power free. It is not known in general whether this will be an infinite sequence, although this has been established for \(k \geq (\sqrt{2}-1/2)d\) by \textit{M. Nair} [Mathematika 23, 159-183 (1976; Zbl 0349.10039)]. The paper studies the distribution of the gaps \(s_{n+1}-s_n\) and proves an asymptotic formula \[ \sum_{n \leq X}(s_{n+1}-s_n)^\gamma \sim B(\gamma,f,k)X \] when the exponent is in the range \(0 \leq \gamma \leq \gamma(k,g)\). The constant \(\gamma(k,g)\) is given explicitly.