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CHAP. V)             GBNEKALIZATIONS OF ETJLEB'S ^-FUNCTION.                  141
where 11(1 -a*) denotes (1 -a*) (1 -&*).. ..
J. Binet161 wrote ih, . . . , ifo ^or ^e integers <N and prime to N=p*q". . . . Then, if Blt -B3, B5, . . . are the Bernoullian numbers 1/6, 1/30, 1/42, . . . ,
s.   i
for a; sufficiently small to insure convergence.   Expanding each member into negative powers of x and comparing coefficients, we pet
the first being equivalent to the usual formula for <KAT).   The general law can be represented symbolically by
where, after expanding the binomials, we are to replace N/(BP) by P_iN and any other term (BP)2*1"1 by jB2A-iP2A-i. It is easily shown that, if k is odd, ST?* is divisible by AT".
Silva26 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 4>iM = Jft<M - This proof of Crelle's result is thus like that by Brennecke.162
W. Brennecke162 proved Crelle's result by means of
Set /x =<t>(n) , a = abc ....    He proved that
the signs being -{-or  according as the number of the distinct prime factors a, 6, ... of n is even or odd.
MlComptes Rendus Paris, 32, 1851, 918-921. 2Programm Realschule, Posen, 1855, 5-6.