CHAP. V]
GENERALIZATIONS or ETJLER'S ^-FUNCTION.
149
these formulas for n = l, 2,..., n, we obtain two determinants of order n, each equal to (~l)n~lJk(n):
1-1-1 0-1 1 ..
1 _2* -3* 0 -5* 6* ..
0 1* 0 -2* 0 -3* ..
00 1* 0 0 -2* ..
lfc 2* 3* 4* ... 1111... 0101... 0010...
L. Gegenbauer204 proved (12). For n=pi"1. . .p/8, set ir(n) = (-
where w(n) denotes the number of distinct prime factors of n. By means of the series f (s) =Sn~8, he proved that, when d ranges over the divisors of r,
the last holding if r has no square factor and following from the third in view of (11),
=0 or
(mn2 == r) ,
according as r is or. is not a square,
where nly . . . , nt range over all sets of solutions of /c = l being due to H. G. Cantor.49
E. Cesaro169 derived (10) from (12), writing f1-fc for Jk.
E. Cesaro205 denoted Jk(ri) by \[/k(n) and gave (12).
L. Gegenbauer170 gave the further generalization
• • • nt+i = n, the case
J. Hammond206 wrote i/'(n, d) for S/(5), where / is an arbitrary function and 5 ranges over all multiples ^ n of the fixed divisor d of n. Then
(13) 2/(«) =^(n, 1) -2^(n, Pl) +S^(n, Plp2) - . . . ,
304Sitzungsber. Ak. Wiss. Wien (Math.), 89 II, 1884, 37-46. Of. p. 841. See Gegenbauer71
of Ch. X.
»08Amiali di Mat., (2), 14, 1886-7, 142-6. ""Messenger Math., 20, 1890-1, 182-190.