A notion of positive definiteness for arithmetical functions (Q2009618)

The authors define a non-negative real function \(f\) on positive integers to be positive definite if for every \(n\) and all sequences \(x_1<x_2<\cdots<x_n\) of positive integers the matrix \[\left[f(\mathrm{GCD}(x_i,x_j))\right]\] is positive semi-definite. They study the properties of such functions and show that by putting \(f\prec g\) if the difference \(g-f\) is positive definite one gets a partial order in the set of all real-valued arithmetical functions on positive integers preserved by multiplication and Dirichlet convolution of positive definite functions. For the entire collection see [Zbl 1426.62010].