Bounds For Determinants Of Matrices Associated With Classes Of Semi-multiplicative Functions

Let f be an arithmetical function and S = {x₁, ...,x_n} be a set of n distinct positive integers. Let (f[x_i, X_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i,j entry. In this paper, we show the following results:Theorem1. Let f be a semi-multiplicative function. Ifthen each of the following is true:(ii). The n × n matrix (f[x_i, x_j]) is positive semi-definite.Theorem2. Let c be a positive integer and f^c denote the c-th Dirichlet convolution of f. For x ∈ Z⁺, we define1/[f^c(x)] := 0 if f^c(x) = 0. If f ∈ C_S and f is a multiplicative function, then each of the following is true:(ii). The n x n matrix (1/{f^c[x_i,x_j]}) is positive semi-definite.