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292                       HlSTOEY OF THE THEOHY OF NUMBERS.                   [CHAP. X
71= oo of T(n)/(n log n) is 1; cf. (7); the mean of SCd+p)-1 is (1+1/2+ . . . +l/p)/p. As generalizations of Berger's51 results, the mean of 2d/pd is l/(p— 1); the mean of the sum of the rth powers of the divisors of n is nr f (r+1) and that of the inverses of their rth powers is f (r+1), where
(12)                               f(s)=Sl/n*.
J. W. L. Glaisher55 proved the last formula of Catalan42 and
<r(n)-er(n-4)-(r(n-8)+<r(n-20)+o-(n-28)- . . .
= Q(n-l)+3Q(n-3)-6Q(n-6)-10Q(n-10)+...,
where Q(n) is the number of partitions of n without repetitions, and 4, 8, 20, ... are the quadruples of the pentagonal numbers. He gave another formula of the latter type.
B,. Lipschitz,56 using his notations,50 proved that
G(n) -
where P ranges over those numbers gn which are composed exclusively of primes other than given primes a, 6, . . . , each rg n.
Ch. Hermite57 proved (11) very simply.
R. Lipschitz58 considered the number rs(t) of those divisors of t which are exact sth powers of integers and proved that
M fni     " rn1/si        „    ^ rni     M r?i1/si = -MX+ S b  + 2 I -IA I = -M2+ S |5| + 2 f-iTi J,
o;=lLXJ       y=lU/      J                a;=lL^J       y = lL.y     J
where p8 is the largest sth power gn, and y = [n//x*]. The last expression, found by taking ju = [n(1+'rl], gives a generalization of (11).
T. J. Stieltjes59 proved (7) by use of definite integrals.
E. Cesaro60 proved (7) arithmetically and (11).
E. Cesaro61 proved that, if d ranges over the divisors of n, and d over those of x,
Taking g(x) = lj(x)=x, <t>(x), l/x, we get the first two formulas of Liouville25
"Messenger Math., 12, 1882-3, 169-170.
"Comptes Rendus Paris, 96, 1883, 327-9.                                                                               «
"Acta Math., 2, 1883, 299-300.
"Ibid., 301-4.
"Comptes Rendus Paris, 96, 1883, 764-6.
*«Ibid., 1029.
"M&n. Soc, Sc. Li^ge, (2), 10, 1883, M<§m. 6, pp. 26-34.