306 History of the Theory of Numbers. [Chap, x 1 C^' 
Ch. Hermite^' proved that if F{N) is the number of odd divisors of N, 
n = l 
and then that 
F(l)+F(2)+ . . . +F{N) =^N log iV+ (c-^N, 
$(l)+$(2)+ . . . +$(iV) =^N log N/k+ (c-^N, 
asATnptotically, where ^(N) is the number of decompositions of N into two 
factors d, d', such that d'>kd. 
E. Catalan^^" noted that, if n = i+i' = 2i"d, 
'La{i)a{i')=2(P, i:{aii)a{2n-i)} =Si:{<j{i)<x{n-i)]. 
E. Ces^ro^ proved Lambert's^ result that T{n) is the coefficient of x" in 
2xV(l — a;*). Let T,{n) be the number of sets of positive integral solutions of 
and s,{n) the sum of the values taken by ^,. Then 
sM = TM + TXn-v) + TXn -2v)+..., 
T{n) =Si(n) -Szin) +83(71) 
Let aa^)=S(-l)'*+^r,(x-n), 
summed for the divisors d of n. Then 
Tin)=tM+t2{n)+ . . .+TM-T2{n) + Ts{n)- . . .. 
E. Busche^^ employed two complementary di\isors 5^ and 8 J of m, 
an arbitrary function/, and a function y=^(x) increasing with x whose 
inverse function is x = ?/)/' (7/). Then 
2 limx)], X) -/(O, x) } =2 {/(5'^, 5 J -/(5'^- 1, 5 J ) , 
x=l 
where in the second member the summation extends over all divisors of all 
positive integers, and $(w)^6;„^a. In particular, 
2 /(x)[tA(x)] =2/(5J, 2 [rA(x)] = number of 5„, 
1=1 2=1 
subject to the same inequalities. In the last equation take \l/(x)=x, 
a = [\/n]', we get (11). 
J. Hacks^® proved that, if 7?i is odd, 
^W^T(l)+r(3)+T(5) + . . . +r(7n) =2[^], 
wjour. fur Math., 99, 1886, 324-8. 
•"^Mdm. .Soc. R. Sc. Li^ge, (2), 13, 1886, 318 (Melanges Math., II). 
•*Jomal de sciencias math, e astr., 7, 1886, 3-6. 
«Jour. fiir Math., 100, 1887, 459-464. Cf. Busche.»" 
"Acta Math., 9, 1887, 177-181. Corrections, Hacks, »^ p. 6, footnote. 
