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306                        HlSTOBY OF THE THEORY OF NUMBERS.                   [CHAP. X
Ch. Hermite93 proved that if F(N) is the number of odd divisors of JV,
and then that
) =|w log N+ c-tf ,
|tf log
asymptotically, where $(N) is the number of decompositions of N into two factors dj d', such that dr>kd.
E. Catalan930 noted that, if n=i+i'=2i"d,
2ff(i)<r(i') =Sd3,           2 {<r(M2n-i) } =8S {<r(i>(n-i) } .
E. Cesaro94 proved Lambert's7 result that r(n) is the coefficient of xn in 2a;Y(l  zfc).   Let ^(n) be the number of sets of positive integral solutions of
and s,(n) the sum of the values taken by .   Then ,(n) = r,
r(n) = ^(
Let                               aaO=S(
summed for the divisors d of n.   Then
E. Busche95 employed two complementary divisors 5m and 5m' of m, an arbitrary function /, and a function j/=$(a;) increasing with x whose inverse function isx = y\l/ (y) . Then
S {/([*(*)], *) ~/(0, *)} -2 {/(' SJ -/('.-!, 5J},
where in the second member the summation extends over all divisors of all positive integers, and <i>(m)^5w^a.   In particular,
,          S [^(x)] = number of dm,
a-l                                            x-1
subject to the same inequalities.   In the last equation take \f/(x)=x} a = [\/n]; we get (11).
J. Hacks96 proved that, if m is odd,
8f(m)sr(l)+T(3)+T(5) + . . .-f-r(m) =
MJour. ftir Math., 99, 1886, 324-8.
93aM6m. Soc. R. Sc. Li^ge, (2), 13, 1886, 318 (Melanges Math., II).
"Jornal de sciencias math, e astr., 7, 1886, 3-6.
96Jour. ftir Math., 100, 1887, 459-464.   Cf. Busche."3
MActa Math., 9, 1887, 177-181.   Corrections, Hacks,07 p. 6, footnote.