294 HISTORY OF THE THEORY OF NUMBERS. [CHAP, x 771 = 2. The mean (pp. 216-9) of the number of divisors of the form a/x+r of nis, forr>0, (cf. pp. 341-2 and, for a = 4, 6, pp. 136-8), while several proofs (also, p. 134) are given of the known result that the number of divisors of n which are multiples of a is in mean If (pp. 291-2) a ranges over the integers for which [2n/d] is odd, the number (sum) of the a's is the excess of the number (sum) of the divisors of n+1, 71+2, . . ., 2n over that of 1, . . . , n; the means are n log 4 and ir2n2/6. If (pp. 294-9) k ranges over the integers for which [n/k] is odd, the number of the k's is the excess of the number of odd divisors of 1, . . ., n over the number of their even divisors, and the sum of the k's is the sum of the odd divisors of 1,. . ., n; also Several asymptotic evaluations by Ces&ro are erroneous. For instance, for the functions X(n) and n(n), denned by Liouville26 and Mertens,37 Ces&ro (p. 307, p. 157) gave as the* mean values 6/Tr2 and 36/Tr4, whereas each is zero.62 J. W. L. Glaisher63 considered the sum A(n) of the odd divisors of n. If n=2rm (m odd), A(n)— cr(m). The following theorems were proved by use of series for elliptic functions : A(l)A(2n-l)+A(3)A(2n-3)+A(5)A(2w-5) + . . . equals the sum of the cubes of those divisors of n whose complementary divisors are odd. The sum of the cubes of all divisors of 2n+l is A(2n+l) + 12{A(l)A(2n)+A(2)A(2n~l)+...+A(2n)A(l)}. If A, B, C are the sums of the cubes of those divisors of 2n which are respectively even, odd, with odd complementary divisor, 2A(2n)+24{A(2)A(2tt-2)+A(4)A(2n-4)+. . . +A(2n-2)A(2)} - l(2A - 2B - C) = i(3-23r - 10) J? <j 7 if 2n = 2rm (m odd). Halphen's formula40 is stated on p. 220. Next, MH. v. Mangoldt, Sitzungsber. Ak. Wiss. Berlin, 1897, 849, 852; E. Landau, Sitzungsber. Ak. Wiss. Wien, 112, Ha, 1903, 537. MQuar. Jour. Math., 19, 1883, 216-223.