292 History of the Theory of Numbers. [Chap, x 
7i= 00 of r(r2)/(n log r?) is l;cf. (7); themean of 2(d+p)-' is (1 + 1/2+... 
+ l/p)/P- -'^s generalizations of Berger's^^ results, the mean of H.d/'p^ is 
l/(p — 1); the mean of the sum of the rth powers of the divisors of n is 
^r ^(/--f 1) and that of the inverses of their rth powers is f(r+l), where 
(12) f(s)=ilM 
n=l 
J. W. L. Glaisher^^ proved the last formula of Catalan^^ and 
(r(n)-(r(n-4)-(r(n-8)+(T(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 tj-pe. 
R. Lipschitz,^^ using his notations,^" proved that 
7'(n)-.r0+.r[^]-...=n+z[|], 
G(n)-SaGg]+2«6G[^] - . . . =n+Sp[|], 
D(.) -SD [2] +XD [^] - . . . =$(n) +2*[|] , 
where P ranges over those numbers ^ n which are composed exclusivelj'' of 
primes other than given primes a,h,. . ., each ^ n. 
Ch. Hermite"^ proved (11) very simply. 
R. Lipschitz^^ considered the number T^it) of those divisors of t which are 
exact sth powers of integers and proved that 
where p' is the largest sth power ^rz, and v = [n/ii']. The last expression, 
found by taking /i = [n^^"^*^" ], gives a generahzation of (11). 
T. J. Stieltjes^^ proved (7) by use of definite integrals. 
E. Cesaro^° proved (7) arithmetically and (11). 
E. Cesaro^^ proved that, if d ranges over the divisors of n, and 5 over 
those of X, 
(13) 2G(d)/Q)=2^(d)FQ), F{x)^i:fi8), G{x)^Xg{8). 
Taking g(x) = l,f{x) =x, 4>{x), 1/x, we get the first two formulas of Liouville^^ 
"Messenger Math., 12, 1882-3, 16&-170. 
"Comptes Rendus Paris, 96, 1883, 327-9. 
"Acta Math., 2, 1883, 299-300. 
"/&id., 301-4. 
5»Cdmpte9 Rendus Paris, 96, 1883, 764-6. 
^Hhid., 1029. 
"Mdm. Soc. Sc. Li^e, (2), 10, 1883, Mem. 6, pp. 26-34. 
