122 HlSTOKY OF THE THEOBY OP NUMBERS. [CHAP. V
the example given being r = 2, n = 4, whence the divisors of n are 1*1, 24, 44, 1-2, 2-2, 14 and the above terms are
M-l, 1-M, M-2, 2-14, 2-1-1, 4-2-1,
with the sum 42. [With this obscure result contrast that by Cantor.49]
G. L. Dirichlet33 completed by induction Euler's2 method of proving (3), obtaining at the same time the generalization that, if p, q, . . . , s are divisors, relatively prime in pairs, of N, the number of integers ^N which are divisible by no one of p, . . . , s is
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. Fontebasso34 repeated the last remark and gave Gauss' proof of (1).
E. Laguerre35 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
where d ranges over the divisors of m. For k = l, this reduces to (4).
F. Mertens36 obtained an asymptotic value for <£(!) + . . . +<I>(G) f°r large. He employed the function ju(n) [see Ch. XIX] and proved that
S 4,(m) = 1 S
m=l n-1
where C is Euler's constant 0.57721 .... This upper limit for A is more exact than that by Dirichlet.21
T. Pepin37 stated that, if n = aab0 . . . (a, &, . . .distinct primes),
Moret-Blanc38 proved the latter by noting that the first sum is the number of integers <n which are divisible by a single one of the prunes a, 6, . . . , the second sum is the number of integers <jn divisible by two of the primes, . . ., while a""1^"1. . . is the number of integers <n divisible by all those primes.
H. J. S. Smith39 considered the w-rowed determinant Am having as the element in the ith row and jth column the g. c. d. (i, j) of i, j. Let l± = ra,
"Zahlentheorie, §11, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894.
34Saggio di una introd. all'arit. trascendente, Treviso, 1867, 23-26.
3SBull. Soc. Math. France, 1, 1872-3, 77.
38 Jour, fur Math., 77, 1874, 289-91.
"Nouv. Ann. Math., (2), 14, 1875, 276.
3S/6wf., p. 374. L. Gegenbauer, Monatsh. Math. Phys., 4, 1893, 184, gave a generalization to
primary complex numbers. "Proc. London Math. Soc., 7, 1875-6, 208-212; Coll. Papers, 2, 161.