120 History of the Theory of Numbers. [Chap, v 
E. Betti^® evaluated 0(w), where 7n is a product of powers of the distinct 
primes ai, a2> • •• • Consider the set Ci of the products of the a's taken i at 
a time and their multiples ^m. Thus Co is 1, . . . , w, while C2 is 
0x02, 20x02, . . ., 0102; cii(h, 2ai03, . . ., diCis',- ■ •• 
aia2 ttitta 
Let X be an integer < w divisible by ai, . . . , a„. Then x occurs 
times in the sets Co, C2, C4, . . . ; and 2" ^ times in Ci, C3, . . .. Summing 
l-(l) + (2)-(3)+ =0 
for each of the m—<t>{m) integers ^m having factors in common with m, 
we get 
m-0(m)-s(j)+s(2)-...=O. 
But ^i-i) is the niunber of integers having in common with m one of the 
factors ai, 02, . . ., and hence equals S— . Next, ^i^j ^^ *^® number of 
integers having in common with m one of the factors 0102, OiOa, . . . , and hence 
equals 2 { m/ (0102) } . Thus 
mm 
4>{m) =m—L, — 1-2 .... 
R. Dedekind^^ gave a general theorem on the inversion of functions (to 
be explained in the chapter on that subject), which for the special case of 
</)(n) becomes a proof like Betti's. Cf. Chrystal's Algebra, II, 1889, 511; 
Mathews' Theory of Numbers, 1892, 5; Borel and Drach,^^ p. 27. 
J. B. Sturm^^ evaluated 4>{N) by a method which will be illustrated for 
the case N = \b. From 1,. . ., 15 delete the five multiples of 3. Among 
the remaining ten numbers there are as many multiples of 5 as there are 
multiples of 5 among the first ten numbers. Hence <^(15) = 10—2 = 8. 
The theorem involved is the following. From the three sets 
1, 2, 3,* 4, 5; 6,* 7, 8, 9,* 10; 11, 12,* 13, 14, 15* 
delete (by marking with an asterisk) the multiples of 3. The numbers 
11, 13, 14 which remain in the final set are congruent modulo 5 to the num- 
bers 6, 3, 9 deleted from the earUer sets. 
J. Liouville" proved by use of (4) that, for |x|<l, 
g <t>{m)x"' _ X (t>{m)x"' _ g <f>{m)x'^ _ x{l-\-x^) 
m^ll-X"^ ~{l-xf l-X^^'m^ll+X"" ~{1-Xy ' 
"Bertrand's Alg^bre, Ital. transl. with notes by Betti, Firenze, 1856, note 5. Proof reproduced 
by Fontebasso'*, pp. 74-77. ^ 
"Jour, fvir Math., 54, 1857, 21. Dirichlet-Dedekind, Zahlentheohe, §138. ^ 
"Archiv Math. Phys., 29, 1857, 448-452. 
"Jour, de math6matiques, (2), 2, 1857, 433^40. 
