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*. n=l 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, (13) 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.