Chap. V] 
Generalizations of Euler's (^-Function. 
149 
these formulas for n = l, 2,. 
each equal to { — lY~'^Jkin): 
1^ 2^ 3^ 4^ 
1111 
10 1 
10 
n, we obtain two determinants of order n, 
1 
-1 
-1 

-1 
1 ... 
1 
-2" 
-3^ 

-5^ 
6^ ... 

1' 

-2' 

-3* . . . 


1* 


-2' ... 
L. Gegenbauer^*^ proved (12). For n = 'pi^ . . .p/'^, set 
7r(n) = (-irpi...p„ X(n) = (-l^+••+^ 
where w{n) denotes the number of distinct prime factors of n. By means 
of the series f (s) =Sn~*, he proved that, when d ranges over the divisors of r, 
ZF(d)d2' = r^ SF(d)d2' = r'SdV,(d), 
^F{d)Jk{d)d'' = 0, S(-l)(^+i)"'(^/V7rM^j =0, 
the last holding if r has no square factor and following from the third in 
view of (11), 
Mr) =2c^m(2) , i:F{d)d'tx{d) =r''fx{r), i:Md)Ud)J,(^^ =0 or J,,{Vr), 
according as r is or is not a square, 
S ( - 1) ^'+^)"'('"V^(m)/fe(m)/2fe(^)w'* = r''\{r)Jk{r) {mn^ = r) , 
TO, n 
^JkM . . .JkinW-^'W-^'" 
.nti=r"', 
where rii, . . ., n^ range over all sets of solutions of nin2. . .n,+i = n, the case 
A; = 1 being due to H. G. Cantor .^^ 
E. Cesaro^^^ derived (10) from (12), writing ^i_k for Jk. 
E. Cesaro^"^ denoted J kin) by xf^'^in) and gave (12). 
L. Gegenbauer^^" gave the further generaUzation 
X{g,{x)f==i: 
i[f]u-), 
[^]. 
J. Hammond^°^ wrote \f/(n, d) for 2/(5), where / is an arbitrary function 
and 5 ranges over all multiples ^ n of the fixed divisor d of n. Then 
(13) mt)=^^P{n, l)-2)/^(n, pi)+2iA(n, p,P2)-..., 
""Sitzungsber. Ak. Wiss. Wien (Math.), 89 II, 1884, 37-46. Cf. p. 841. See Gegenbauer" 
of Ch. X. 
"'Annali di Mat., (2), 14, 1886-7, 142-6; 
"'Messenger Math., 20, 1890-1, 182-190. 
