A Turán-Kubilius inequality for integer matrices (Q1281636)

The authors prove a general Turán-Kubilius inequality and use it to derive that the number \(\tau(S)\) of divisors of an integer \(r\times r\) matrix \(S\) verifies \(\tau(S)= (\log|S|)^{\log 2+ o(1)}\) for all but \(o(X)\) matrices of determinant \(\leq X\). This is in sharp contrast with the average order which is \(\asymp|S|^{\beta_r-1} (\log |S|)^{\gamma_r}\), for \(\beta_r\) that are \(>1\) as soon as \(r\geq 4\) and some nonnegative \(\gamma_r\).