Chap. V] 
Euler's 0-Function. 
137 
G. A. Miller^"^ proved (4) by noting that in a cyclic group G of order N 
there is a single cyclic subgroup of order d, a divisor of N, and it contains 
0((i) operators of order d, while the order of any operator of G is a divisor 
of N. Thus (4) states merely that the order of G equals the sum of the 
numbers of the operators of the various possible orders. Next, (1) follows 
from an enumeration of the operators of highest period AB in a cyclic group 
of order AB, which is the direct product of its cyclic subgroups of orders 
A and B. Finally, if p is a prime, all the subgroups of a cyclic group of 
order p" are contained in its subgroup of order p"~\ whence <^(p") = p" — p"~^ 
G. A. Miller^°^ proved the last three theorems and the fact that 0(0 is 
even if Z>2 by means of the properties of the abelian group whose elements 
are the integers <m which have with m a g. c. d. equal to k. 
K. P. Nordlund^"^ proved 4){mn ...) = (m — l)(n — 1)..., where m, n,. . . 
are distinct primes, by writing down the multiples <mnp of m, the multi- 
ples of mn, etc., whence the number of integers Kmnp and not prime to it 
is mnp — l — {m — l){n — l){p — l), 
E. Busche^*^ treated geometrically systems (td) of four integers such 
that ad — hc>0, evaluated the number $(aS) of systems incongruent modulo 
S and prime to S, and generalized (4) to 2$(*S). 
L. Orlando^°^ showed that 0(m) is determined by (4) [Lucas''^]. 
H. Bonse^°^ proved Maillet's^^ theorem for r = l, 2, 3 without using 
•■Tcheby chef's theorem. His lemma was generalized by T. Suzuki. ^°^" 
J. Sommer^^^ gave without reference Crelle's^ final evaluation of (/)(n). 
R. D. CarmichaeP*'^ proved that if n is such that (l){x)=n is solvable 
there are at least two solutions x. He found solutions of </)(x) = 2" [in accord 
with Pichler^^] and proved that there are just n+2 solutions (a single one 
being odd) when n^31 and just 33 solutions when 32^ n^ 255. All the 
solutions of <^(x) = 4n — 2> 2 are of the form p", 2p", where p is a prime of the 
form 4s — 1 ; for example, if n = 5, the solutions are 19, 27 and their doubles. 
CarmichaeP°® gave a table showing every value of m for which 0(m) 
has any given value ^ 1000. 
A. Ranum^^^" would solve 4>{x) = n by resolving n in every possible way 
into factors no, ., n^, capable of being taken as the values of 0(2*"), 4>{pi'), 
. . ., 0(pA)) where 2, pi, . . ., p, are distinct primes. Then 2'^pi"'. . .p^^'r is 
a value of x. 
CarmichaeP^" gave a method of solving (j>(x)=a, based on the testing 
of the equation for each factor x of a definite function of a. 
M. Fekete^^^ considered the determinant pkn obtained by deleting the 
last row and last column of Sylvester's eliminant for a;''' — 1 = and a;** — 1 = 
"lAmer. Math. Monthly, 12, 1905, 41-43. 
"'Amer. Jour. Math., 27, 1905, 315. 
"'Nyt Tidsskrift for Mat., 16A, 1905, 15-29. 
iMJour. fiir Math., 131, 1906, 113-135. 
"»Periodico di Mat., 22, 1907, 134-6. 
iwArchiv Math. Phya., (3), 12, 1907, 292-5. 
i^^Tohoku Math. Jour., 3, 1913, 83-6. 
"'Vorlesungen liber Zahlentheorie, 1907, 5. 
"SBull. Amer. Math. Soc, 13, 1907, 241-3. 
"»Amer. Jour. Math., 30, 1908, 394-400. 
lo'^^Trans. Amer. Math. Soc, 9, 1908, 193-4. 
"«BuU. Amer. Math. Soc, 15, 1909, 223. 
"iMath. 6s Phys. Lapok (Math. Phys. Soc), 
Budapest, 18, 1909, 349-370. German 
transl., Math. Naturwiss. Berichte aus 
Ungam, 26, 1913 (1908), 196. 
