On the probability that the values of m polynomials have a given g.c.d
Let \(m\geq 2\), \(h\geq 1\) be integers and \(f_ 1,...,f_ m\) be arbitrary polynomials with integer coefficients and of positive degree. The authors derive an asymptotic formula with an error term for \(M(x;f_ 1,...,f_ m;h),\) the number of m-tuples \(<x_ 1,...,x_ m>\) of positive integers such that \(g.c.d.(f_ 1(x_ 1),...,f_ m(x_ m))=h\) and each \(x_ i\leq x\), and then go on to study the density \(d(f_ 1,...,f_ m;h)=\lim_{x\to \infty}x^{-m} M(x;f_ 1,...,f_ m;h),\) obtaining conditions under which it vanishes and giving some examples to illustrate their results. They point out that there are errors in \textit{R. N. Buttsworth}'s paper [ibid. 12, 487-498 (1980; Zbl 0443.10001)] on the evaluation of \(d(f_ 1,...,f_ m;h)\). For the case \(f_ i(x)=x\) \((i=1,...,m)\) and \(h=1\), it was shown by \textit{J. E. Nymann} [ibid. 4, 469-473 (1972; Zbl 0246.10038)] that the corresponding density is 1/\(\zeta\) (m).
scientific article; zbMATH DE number 4002211