On the Sum Σ (a-1, n), [(a, n) = 1]

The object of this paper is to establish a class of identities of which a typical one is

(l-l)

Σ (a-l,n ) = φ(n) d(n),

(a, n) = 1

where (m, n) denotes the g.e.d. of m and n,φ ¦n) is the number of numbers less than n and prime to n, d(n) is the number of divisors of n and the summition is over all residues a mod n that are prime to n.