Chap. X] SuM AND NuMBER OF DiVISORS. 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. Lucas^^ 
o-(l) + . . . +o-(m) =0(1, m) + . . . +0(m, m). 
J. W. L. Glaisher^^ stated that if f{n) is the sum of the odd divisors of n 
and if g{n) is the sum of the even divisors of n, and /(O) =0, g{0) =n, then 
/(n)+/(n-l)+/(n-3)+/(n-6)+/(n-10) + ... 
= g{n)+g{n-l)+gin-S)+ . . .. 
Chr.Zeller^^ proved (11). 
R. Lipschitz^o wrote G(t) for <t{1)-\- . . .-\-a{t), D{t) for {t''+t)/2, and 
$(0 for (^(1) + . . . +(f>{t), using Euler's (f>{t). Then if 2, 3, 5, 6, . . . are the 
integers not divisible by a square > 1 , 
^w--[l]--[l]--G] 
+ . . . =n, 
G(n)-2G 
D(n)-D 
the sign depending on the number of prime factors of the denominator. He 
discussed (pp. 985-7) Dirichlet's^'^ results on the mean of T{n), o-(n), (f>{n). 
A. Berger^^ proved by use of gamma functions that the mean of the sum 
of the divisors d of n is ir^n/Q, that of S d/2'^ is 1, that of Sl/d! is ir^/Q, 
G. Cantor^^" gave the second formula of Liouville^^ and his^^ third. 
A. Piltz^^ considered the, number Tk{n) of sets of positive integral solu- 
tions of Ui. . .Uk = n, where differently arranged u's give different sets. 
Thus T'i(n) = 1, T'2(n) =T{n). If a- is the real part of the complex number s, 
and n* denotes e^ '°^ " for the real value of the logarithm, he proved that 
n=l "' m = 
where l = \—(T — \/k, and the 6's are constants, 6^ = for s ?^ 1 ; while 0(/) is^'^ 
of the order of magnitude of /. Taking s = 0, we obtain the number 'SiTkin) 
of sets of positive integral solutions of t^i . . .u^'^x. 
H. Ahlborn^Hreated (11). 
E. Cesaro^^ 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 
^^Nouv. Corresp. Math., 5, 1879, 296. 
48NOUV. Corresp. Math., 5, 1879, 176. 
"Gottingen Nachrichten, 1879, 265. 
^oComptes Rendus Paris, 89, 1879, 948-50. Cf. Bachmann^" of Ch. XIX. 
"Nova Acta Soc. Sc. Upsal., (3), 11, 1883, No. 1 (1880). Extract by Catalan in Nouv. Corresp. 
Math., 6, 1880, 551-2. Cf. Gram.«^« 
"«G6ttmgen Nachr., 1880, 161; Math. Ann., 16, 1880, 586. 
"Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natiirlichen Zahlen ala 
Produkte einer gegebenen Anzahl Faktoren mit der Grosse der Zahlen wachst. Diss., 
Berlin, 1881. 
"Progr., Hamburg, 1881. 
"Mathesis, 1, 1881, 99-102. Nouv. Ann. Math., (3), 1, 1882, 240; 2, 1883, 239, 240. Also 
Ces^ro," 113-123, 133. 
