Let f(x) be an increasing function. Recently there have been several papers which proved that under fairly general conditions on f(x) the density of integers n for which (n, f(n)) = 1 is 6/π2 and that (d(n) denotes the number of divisors of n)

In particular both of these results hold if f(x) = xα, 0 < α < 1 and the first holds if f(x) = [α x], α irrational.

The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).

A partial homomorphism ϕ from B into A will be called a fractional homomorphism if dom ϕ ≤ B (A). Every extension of a fractional homomorphism is again fractional, and each fractional homomorphism has a unique irreducible extension.

Given an integer n, we define an r-composition of n as follows:

An r-composition of n, (t1, ... , tr), is a set of ti where ti ≥ 1 is an integer for i = 1, ... , r such that

If r is an integer such that 1 ≤ r ≤ n, we have, obviously, distinct r - compositions of n.

Let x, y be column matrices of n real homogeneous coordinates xj, yj (j = 1,..., n) representing points in (n - 1) - dimensional real projective space Pn - 1. Let A be an n × n real symmetric matrix. The equation x' Ax = 0 represents a quadric in Pn - 1 and the equation

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.

Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.

Let x, y denote real column vectors with n elements. Let A be a regular symmetric real n×n matrix. Dashes indicate transposition.

If x is fixed, x' Ax > 0, the discriminant of A at x is the quadratic form

y' where S= S(x)= x'x'Ax ¨ Axx'A.

In Can. Math. Bull. 1, pp. 175-179, Dr. Schwerdtfeger proved the equivalence of the following properties of A:

1. (i) A is of the congruence type [+, -,..., -].

2. (ii) y' Sy≤ 0 for all y, equality holding if and only if y is a multiple of x. His note is of particular interest because he also discusses the eigen-values of S. If only the quoted result is aimed at, the following procedure may be shorter.

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equation

so that there corresponds the matrix B to the bilinear form

4.1

on the linear space

4.2

and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.

The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.