A Turán–Kubilius Inequality for Integer Matrices Original Research Article

We prove a general Turán–Kubilius inequality and use it to derive that the numberτ(S) of divisors of an integerr×rmatrixSverifiesτ(S)=(Log S)Log 2+o(1)for all buto(X) matrices of determinant less-than-or-equals, slantX. This is in sharp contrast with the average order which is asymptotically equal toSβr−1(Log S)γrforβrthat are >1 as soon asrgreater-or-equal, slanted4 and some non-negativeγr. We further extract a fairly large set of matrices over which the normal order is much closer to the average order.