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CHAP. X]                         SUM AND  NuMBBK OF  DlVISOBS.                                291
Radicke (p. 280) gave an easy proof and noted that if we take n = 1, . . . , m and add, we get the result by E. Lucas47
J. W. L. Glaisher48 stated that if f(n) is the sum of the odd divisors of n and if g(ri) is the sum of the even divisors of n, and /(O) =0, g(0) =n, then
Chr.Zeller49 proved (11).
R. Lipschitz50 wrote 0(0 for <r(l)+. . .+cr(0, D(t) for (*2+*)/2, and for <K1) + v . +*(0, using Euler's 4>(t).   Then if 2, 3, 5, 6, ... are the integers not divisible by a square > 1,
the sign depending on the number of prime factors of the denominator. He discussed (pp. 985-7) Dirichlet's17 results on the mean of r(n), o*(n), $(n).
A. Berger51 proved by use of gamma functions that the mean of the sum of the divisors d of n is 7r2n/6, that of S d/2d is 1, that of Sl/d is 7r2/6.
G. Cantor510 gave the second formula of Liouville25 and his26 third.
A. Piltz52 considered the number Tk(ri) of sets of positive integral solutions of Ui. . .uk = n, where differently arranged u's give different sets. Thus jTj(n) = 1, T2(n)=r(n). If a is the real part of the complex number s, and n* denotes e* log n for the real value of the logarithm, he proved that
tk(x] s) = I ^jM = ^-a S &mlo
where Z = 1 - <r - 1 /k, and the fr's are constants, bk = 0 f or s ^ 1 ; while 0 (/) is90 of the order of magnitude of /. Taking s = 0, we obtain the number 1,Tk(ri) of sets of positive integral solutions of Ui . . .uk^x.
H. Ahlborn63 treated (11).
E. Cesaro64 noted that the mean of the difference between the number of odd and number of even divisors of any integer is log 2 ; the limit for
47Nouv. Corrcsp. Math., 5, 1879, 296.
"Nouv. Corresp. Math., 5, 1879, 176.
49G6ttingen Nachrichten, 1879, 265.
"Comptes Rendus Paris, 89, 1879, 948-50.   Cf. Bachmann20 of Ch. XIX.
6lNova Acta Soc. Sc. Upsal., (3), 11, 1883, No. 1 (1880).  Extract by Catalan in Nouv. Corresp.
Math., 6, 1880, 551-2.    Cf. Gram.040 BlaG6ttingen Nachr., 1880, 161; Math. Ann., 16, 1880, 586.      MUeber das Gesetz, nach wdchem die mittlere Darstellbarkeit der natiirlichen Zahlen ala
Produkte einer gegebenen Anzahl Faktoren mit der Grosse der Zahlen wachst.   Diss.,
Berlin, 1881. MProgr., Hamburg, 1881. "Mathesis, 1, 1881, 99-102.    Nouv. Ann. Math., (3), 1, 1882, 240; 2, 1883, 239, 240.    Also
Cesaro,61 113-123, 133.