A TURAN-KUBILIUS INEQUALITY FOR INTEGER MATRICES

We prove a general Turan-Kubilius inequality and use it to derive that the number tau(S) of divisors of an integer r x r matrix S verifies t au(S) = (Log \S\)(Log 2 + o(1)) for all but o(X) matrices of determina nt less than or equal to X. This is in sharp contrast with the average order which is asymptotic to[S](beta r-1)(Log \S\)(gamma r) for beta( r) that are >1 as soon as r greater than or equal to 4 and some non-ne gative gamma(r). We further extract a fairly large set of matrices ove r which the normal order is much closer to the average order. (C) 1998 Academic Press.