Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime

Abstract

Let $F=(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \[ {\gcd}_F(x_1,\ldots,x_k)= \gcd(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k)). \] We obtain an unconditional asymptotic formula for the sum \[ \sum_{1\le x_1,\ldots,x_k\le x} h({\gcd}_F(x_1,\ldots,x_k)), \] where $F$ is a system of $m\ge 2$ polynomials of $k\ge 2$ variables subject to certain general properties, and $h$ is a bounded strongly multiplicative function. In particular, we deduce an asymptotic formula with error term concerning the density of $k$-tuples of positive integers $(x_1,\ldots,x_k)$ such that ${\gcd}_F(x_1,\ldots,x_k)=1$, given by the Ekedahl-Poonen formula.