Chap. V] EuLER's (^-FUNCTION. 117 
J. A. Grunert^^ examined in a very elementary way the sets 
jk+1, jk-{-2,..., jk+k-l, {j+l)k (j = 0, l,...,p-l) 
and proved that 4){'pk)='p4){k) if the prime p divides k, while 4){pk) = 
(p — l)<j){k) if the prime p does not divide k. From these results, (2) is 
easily deduced [cf. Crelle^^ on (f){Z)]. 
L. Poinsot^® gave Catalan's^^ proof of (4) and proved the statements 
made by Euler^ in his proof of (3) . Thus to show that, of the N' = N{1- 1/p) 
integers < N and prime to p, exactly N'/q are divisible by q, note that the 
set 1,. . ., N contains N/q multiples of q and the set p, 2p, . . . contains 
{N/p)/q multiples of q, while the difference is N'/q. 
If P, Q, R,. . . are relatively prime in pairs, any number prime to 
N = PQR . . . can be expressed in the form 
pQR...+qPR...+rPQ... + ..., 
where p is prime to P, q to Q, etc. If also p<P, q<Q, etc., no two of these 
sums are equal. Thus there are 0(P)0(Q) . . . such sums [certain of which 
may exceed N]. 
To prove (4), take (pp. 70-71) a prime p of the form kN+l and any one 
of the N roots p of a;^= 1 (mod p). Then there is a least integer d, sl divisor 
of N, such that p'^= 1 (mod p). The latter has (f)(d) such roots. Also p is a 
primitive root of the last congruence and of no other such congruence whose 
degree is a divisor of N. 
A. L. Crelle^^ considered the product E = eie2. . .e„ of integers relatively 
prime in pairs, and set Ej = E/ej. When x ranges over the values 1, . . ., Ci, 
the least positive residue modulo E of EiXi-\- . . . +£'„a;„ takes each of the 
values 1, . . .,E once and but once. In case Xi is prime to ei for i = 1, . . . , n, 
the residue of SE'^Xi is prime to E and conversely. Let dn, di2, ... be any 
chosen divisors >1 of e^ which are relatively prime in pairs. Let \}/{ei) 
denote the number of integers ^e^ which are divisible by no one of the 
^ti, di2,. . .. Let yl/{E) be the number of integers ^E which are divisible 
by no one of the dn, di2, c^2ij ■ ■ •> including now all the d's. Then \1/{E) = 
^(ei) . . . i/'(en). In case dn, di2, . . . include all the prime divisors > 1 of e,-, 
ypie^ becomes ^(e^). Of the two proofs (pp. 69-73), one is based on the 
j&rst result quoted, while the other is like that by Gauss .^ 
As before, let ^{y) be the number of integers '^y which are divisible by 
no one of certain chosen relatively prime divisors di,...,dm of y. By con- 
sidering the xy numbers ny-\-r (0^n<x, I'^r^y), it is proved (p. 74) that, 
when X and y are relatively prime, 
ypixy) =x\p{y), \p2ixy) = {x-l)xl/{y), 
where \p2{^y) is the number of integers ^xy which are divisible neither by 
X nor by any one of the d's. These formulas lead (pp. 79-83) to the value 
of0(Z). Set 
Z = p/'...p/M, z = Pi...p^, n = Z/z, 
i^Archiv. Math. Phys., 3, 1843, 196-203. 
"Jour, de Math^matiques, 10, 1845, 37-43. 
"Encyklopadie der Zahlentheorie, Jour, fiir Math., 29, 1845, 58-95. 
