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314                  HISTORY OF THE THEOKY OF NUMBEES.              [CHAP, x
summed for p, <r =0, 1, . . ., with p^cr.   Also,
l _ 1
-+2S (-1)"
where G'(&) is the number of odd divisors of k; \l/'(n, a) is the number of divisors >a of n whose complementary divisors are odd; while ^(k, ju) is the number of even divisors >M of fc,
In No. 33, he expressed in terms of greatest integer functions
S{^(w p -OTI, &+<r)~x(w~-p--<>^)}>
E. Busche123 gave a geometrical proof of MeisseFs22 (11). J. Schroder124 obtained (11) and the first relation (15) of Lerch98 as special cases of the theorem that
0,1,2,...                   m          m
5)   ^(n-rS ip
Pi-..., Pin-                       -l          t
equals the coejfficient of xn hi the expansion of
where \l/rv+9(o<) ft) is the number of divisors of a which are >@ and have a complementary divisor of the form rv+s(v = 0, 1, . . .).   He obtained
Schroder125 determined the mentioned coefficient of xn. Schroder126 proved the generalization of (11):
pi Lp J     P=2 For <r(l) +    +<r(ri), Dirichlet,17 end, he gave the value
E. Busche127 proved that if X=$(m) is an increasing (or decreasing) function whose inverse function is m=i(T), the divisors of the natural
1MMittheilungen Math. Gesell. Hamburg, 3, 1894, 167-172. Ibid.t 177-188. >76ttf.f 3, 1897, 302-8. /&ui., 3, 1895, 219-223. lbid.t 3, 1896, 239-40.