On the distribution of inverses modulo n

Let n > 2 be an integer, and for each integer 0 < a < n with (a, n) = 1, define (a) over bar by the congruence a (a) over bar = 1 (mod n) and 0 < (a) over bar < n. The main purpose of this paper is to study the distribution behaviour of \a - (a) over bar\, and prove that for any fixed positive number 0 < delta less than or equal to 1, lim # {a: 1 less than or equal to a less than or equal to n - 1, (a, n) = 1, \a - (a) over bar\ < delta n}/phi(n) = delta(2 - delta), n --> infinity where phi(n) is the Euler function, and # {...} denotes the number of elements of the set {...}. (C) 1996 Academic Press, Inc.