Calculating a determinant associated with multiplicative functions.

Let h be a complex valued multiplicative function. For any natural N, we compute the value DN:= det (h((i,j))/ij) with i|N,j|N, where (i,j) denotes the greatest common divisor of i and j, which appear in increasing order in rows and columns. We prove that the 1/t(N) root of DN (where t(N) is the number of divisors of N) is a multiplicative function of N. The algebric apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions f(n), with 0<=f(p)<1, as minimal values of certain quadratic forms on the t(N) units sphere. The second one is the explicit evaluation of the minimal values of certain others quadratic forms also on the units sphere.