Calculating a determinant associated with multiplicative functions

Codecá, P. ; Nair, M.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 545-555

Résumé

Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_{N}:= \det _{i|N, j|N}\left(\frac{h((i,j))}{ij} \right)$ where $(i, j)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_{N}= \prod _{p^{l}\| N}\left(\frac{1}{p^{l(l+1)}}\prod_{i=1}^{l}(h(p^{i})-h(p^{i-1})) \right)^{\tau (N/p^{l})}.$$ This means that $D_{N}^{1/\tau(N)}$ is a multiplicative function of $N$. The algebraic 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 $0\leq f (p)1$, as minimal values of certain quadratic forms on the $\tau(N)$ unit sphere. The second one is the explicit evaluation of the minimal values of certain others quadratic forms also on the unit sphere.