We prove a general Tura n Kubilius inequality and use it to derive that the number {(S) of divisors of an integer r_r matrix S verifies {(S)=(Log |S|) Log 2+o(1) for all but o(X) matrices of determinant X. This is in sharp contrast with the average order which is Ä |S| ; r &1 (Log |S|) # r for ; r that are >1 as soon as r 4 and some non-negative # r . We further extract a fairly large set of matrices over which the normal order is much closer to the average order.
1998 Academic Press
I. INTRODUCTION
In 1934 1936, Tura n devised a simple but powerful process of obtaining normal orders for some additive functions of integers, a process which was later (1956 1964) extended by Kubilius to every additive function of integers and which is now called the Tura n Kubilius inequality (A classical introduction on this subject may be found in [El].) The simplicity of this proof enables one to extend it to different situations: Horadam [Ho] for instance extended it to some additive functions over a set of regular Beurling generalized integers, while Hinz [Hi] recently extended it to additive functions of integer ideals of a number field. In Section III we describe a general setting which covers these three applications as well as the two new ones that have in fact been the starting point of this paper. The hypotheses are expected to be wide enough to cover most results of this kind. Note, however, that our proof resembles Tura n's.
Our first aim is to study the distribution of the number of divisors of non-singular integer r_r matrices, where r 2. We denote the set of such matrices by Inv r . Since the number of divisors of S, which we shall denote by {(S), depends only on the Smith Normal Form (SNF) of S we study the distribution of the values of {(S) when S ranges over all SNF matrices with determinant atmost X, where X is a large enough real parameter. We recall that the cardinality of such a set is asymptotic to C(r) X (cf. [B1]) where Article No. NT982281