122 History of the Theory of Numbers. [Chap, v 
the example given being r = 2, n = 4, whence the divisors of n are 1-1, 2-1, 4-1, 
1-2, 2-2, 1-4 and the above terms are 
Ml, Ml, M-2, 2-M, 2M, 4-21, 
with the sum 4". [With this obscure result contrast that by Cantor/^] 
G. L. Dirichlet^^ completed by induction Euler's^ method of proving (3), 
obtaining at the same time the generalization that, if p, g, . . . , s are divisors, 
relatively prime in pairs, of N, the number of integers ^ N which are divi- 
sible by no one of p, . . . , s is 
H-;)04) 0--.) 
A proof (§13) of (4) follows from the fact that, if d is a divisor of N, there 
are exactly </)(d) integers ^N having with N the g. c. d. N/d. 
P. A. Fontebasso^^ repeated the last remark and gave Gauss' proof of (1). 
E. Laguerre^^ employed any real number k and integer m and wrote 
(m, m/k) for the number of integers ^m/k which are prime to m. By 
continuous variation of k he proved that 
i:{d,d/k) = [m/k], 
where d ranges over the divisors of m. For k = l, this reduces to (4). 
F. Mertens^^ obtained an asymptotic value for 0(1)+ . . . +(i>iG) for G 
large. He employed the function ju(n) [see Ch. XIX] and proved that 
I 0(m) = | S /.(n){r^lVr^l U^GHA 
TO=i n=i iLnJ Lnjj tt 
|A|<G(ilog,G+iC+f) + l, 
where C is Euler's constant 0.57721 .... This upper limit for A is more 
exact than that by Dirichlet.^^ 
T. Pepin^'' stated that, if n = a"6^. . . (a, 6, . . .distinct primes), 
n=0(n)+2a»-V(^) +2a-V-^(/,(^) + . . . -\-a-'b'-'. . .. 
Moret-Blanc^^ proved the latter by noting that the first sum is the num- 
ber of integers < n which are divisible by a single one of the primes a, 6, ... , 
the second sum is the number of integers < n divisible by two of the primes, 
. . ., while a''~^6^~\ . . is the number of integers <n divisible by all those 
primes. 
H. J. S. Smith^^ considered the m-rowed determinant A„, having as the 
element in the ith. row and jth column the g. c. d. {i, j) of i, j. Let li = m, 
"Zahlentheorie, §11, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 
»*Saggio di una introd. all'arit. trascendente, Treviso, 1867, 23-26. 
«BuU. Soc. Math. France, 1, 1872-3, 77. 
»«Jour. fiir Math., 77, 1874, 289-91. 
»'Nouv. Ann. Math., (2), 14, 1875, 276. 
"/bid., p. 374. L. Gegenbauer, Monatsh. Math. Phys., 4, 1893, 184, gave a generahzation to 
primary complex numbers. 
»»Proc. London Math. Soc, 7, 1875-6, 208-212; CoU. Papers, 2, 161. 
