314 History of the Theory of Numbers. [Chap, x 
summed for p, o- = 0, 1, . . ., with p^<r. Also, 
"s (-l)V(w-a,a)=2 2 (-l)"e'(/n-a) + (-l)'"m, 
a-O a=0 
m— 1 m PtwH 
2^'(7n-a,a)=2(-l)*-Mf , 
a-O t = l L^J 
2 Um-a, 2a) =m+^-ii- ^+2 2 (-1)' "^ , 
«=o ^ v=i L if J 
where 6' (A") is the number of odd divisors of k; yp'{n, a) is the number of 
div'isors >a of n whose complementary di\isors are odd; while \po{k, n) is the 
number of even di\'isors >/i of k. 
In No. 33, he expressed in terms of greatest integer functions 
X{\p{7n—p—an, k-jr(T)—xi'm—p — (7n,n)}} 
2{^(w — a, k-\-a) — {k-]-a)\l/{m — a, k-\-a)}- 
a 
E. Busche^^^ gave a geometrical proof of Meissel's^^ (11). 
J. Schroder^^ obtained (11) and the first relation (15) of Lerch^^ as 
special cases of the theorem that 
0,1,2,... m m 
2 \J/r,+sin-ri: ipi, 2p,) 
Pi.--.Pm= » = 1 » = 1 
equals the coefficient of x" in the expansion of 
m-l 
1- n (l-a:"+0 
1=0 
li::ri ' 
(l-x^"*) n(l-x"+0 
t=0 
where ypr,+>{o-j ^) is the number of di\isors of a which are >^ and have a 
complementary divisor of the form rv-{-s{v = 0,l,. . .). He obtained 
2 \f/r,+i{n-rp, p) = y J • 
Schroder^^ determined the mentioned coefficient of x". 
Schroder ^^^ proved the generaUzation of (11): 
p=iLpJ p=iLpJ p=2 LpJ 
For (j{\)-\- . . . +(r(n), Dirichlet," end, he gave the value 
E. Busche^^^ proved that if X = 4)(m) is an increasing (or decreasing) 
function whose inverse function is m=<l>(X), the divisors of the natural 
i»Mittheilungen Math. GeseU. Hamburg, 3, 1894, 167-172. 
"*Ibid., 177-188. 
^Ibid., 3, 1897, 302-8. 
"•/feid., 3, 1895, 219-223. 
"Ubid., 3, 1896, 239-40. 
