Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 141 
wheren(l-aO denotes {l-a^){l-h'). . .. 
J. Binet^" wrote Vif ■ ■> Vn for the integers <iV and prime to N^p^q". . .. 
Then, if B^, —B^, B^,... are the BernouUian numbers 1/6. 1/30, 1/42, . . ., 
andP,= (l-p^)(l-g'')..., 
for X sufficiently small to insure convergence. Expanding each member into 
negative powers of x and comparing coefficients, we get 
n =277/ = P_,N, 2Sr7i = P_,N\ SXrjf = P_,N^+SB,P,N, 
^n,^ = P_,N^+QB,P,N^.. . 
the first being equivalent to the usual formula for 0(iV). The general law 
can be represented symbolically by 
givr'=^\{N+Bpy-h{N-Bpy\, 
where, after expanding the binomials, we are to replace N"/{BP) by P^iN" 
and any other term {BPY^~'^ by B2h-\P2h-\- It is easily shown that, if k is 
odd, Hit]'' is divisible by N. 
Silva^^ used his symbolic formula, taking S to be the sum of 1, . . ., n, 
whence S{a) is the sum §n(l+n/A) of the multiples ^n of A. Thus 
^i(^) = 2^(^) • This proof of Crelle's result is thus like that by Brennecke.^" 
W. Brennecke^^^ proved Crelle's result by means of 
H-...+n-la(l+2+...+^)+6(l+...+^) + ...t 
+ ]4+...+;J + . ..! + .... 
Set )Li = 0(n) , a = ahc .... He proved that 
<i>^{n)=^}xn'-^\aixn^-^n{\-a^){\-h^) . . ., 
the signs being + or — according as the number of the distinct prime 
factors a, 6, . . . of n is even or odd. 
•"Comptes Rendus Paris, 32, 1851, 918-921. 
"^Programm Realschule, Posen, 1855, §§5-6. 
