294 History of the Theory of Numbers. [Chap, x 
m = 2. The mean (pp. 216-9) of the number of divisors of the form aix+r 
of n is, for r>0, 
i+ijlog«/a+2C-/;ij^dx} 
(cf. pp. 341-2 and, for a = 4, 6, pp. 136-8), while several proofs (also, p. 134) 
are given of the known result that the number of divisors of n which are 
multiples of a is in mean 
-(log7i/a+2C). 
a 
If (pp. 291-2) a ranges over the integers for which [2n/d] is odd, the 
number (sum) of the a's is the excess of the number (sum) of the divisors of 
n + 1, n+2, . . . , 2n over that of 1, . . . , n; the means are n log 4 and 7rW/6. 
If (pp. 294-9) k ranges over the integers for which [n/k] is odd, the number 
of the A:'s is the excess of the number of odd divisors of 1, . . ., n over the 
number of their even divisors, and the sum of the A;'s is the sum of the odd 
divisors of 1, . . . , w; also 
S*W=9^ 9=[^]' 
Several asymptotic evaluations by Cesaro are erroneous. For instance, 
for the functions \{n) and At(n), defined by Liouville^^ and Mertens,^^ 
Cesaro (p. 307, p. 157) gave as the mean values 6/7r^ and 36/7r*, whereas 
each is zero.^^ 
J. W. L. Glaisher^^ considered the sum A(n) of the odd divisors of n. 
If n = 2^m {m odd), A(n) =(j{m). The following theorems were proved by 
use of series for elliptic functions : 
A(l)A(2n-l)+A(3)A(2n-3)+A(5)A(2n-5)+...+A(2n-l)A(l) 
equals the sum of the cubes of those divisors of n whose complementary 
divisors are odd. The sum of the cubes of all divisors of 2n+l is 
A(2n+l) + 12{A(l)A(2n)+A(2)A(2n-l)+. . . +A(2n)A(l)). 
If A, £, C are the sums of the cubes of those divisors of 2n which are respec- 
tively even, odd, with odd complementary divisor, 
2A(2n)+24JA(2)A(2n-2)+A(4)A(2n-4)+. . .+A(2n-2)A(2)) 
= i(2A-2J5-C)=i(3-23^-10)5 
o 7 
if 2n = 2'"m (ttz odd). Halphen's formula^*' is stated on p. 220. Next, 
n(r(2n+l) + (n-5)(r(2n-l) + (n-15)(T(2n-5) 
+ (n-30)(7(2n-ll)+. . . =0, 
"H. V. Mangoldt, Sitzungsber. Ak. Wiss. Berlin, 1897, 849, 852; E. Landau, Sitzungsber, Ak. 
Wiss. Wien, 112, II a, 1903, 537. 
«Quar. Jour. Math., 19, 1883, 216-223. 
