Abstract/Summary

We study the asymptotic behaviour of the singular values of matrices with entries $a_{ij}=f(i/j)$ if $j|i$ and zero otherwise, with $f$ an arithmetical function. In particular, we study the case where $f$ is multiplicative and $F(x):=\sum_{n\leq x} |f(n)|^2$ is regularly varying. Our main result is that, under quite general conditions, the singular values are, asymptotically, $\sqrt{\mu_r F(n)}$, where $\{\mu_r:r=1,2,3,\ldots\}$ are the eigenvalues of some positive Hilbert-Schmidt operator.