444 History of the Theory of Numbers. [Chap, xix 
Those di\'isible by Oi and 02, but by no one of 03, ... , a^, form the sub-set 
/(oa,. . ., a J. Finally, those divisible by fli. . ., O;. form the sub-set desig- 
nated /(O). Thus 
F{ai,a-2, . . ., Ok) =/(ai, Oo, . . . , a^) +2/(a2, as, •• • , Oa) +|/(a3, ^4, • • • , a^t) 
+ ...-f2/(a,)H-/(0), 
7»— 1 
where S indicates that the summation extends over all combinations of 
Oi,. . . , flr taken* /c — r at a time. 
WTien we have any such set or function /(oi, . . . , a^t), uniquely determined 
by Oi, . . . , Qk, independently of their order, and we define F by the foregoing 
formula, then we have the inverse formula 
/(oi, 02, . . . , a„) =F(ai, 03, ... , aj -2/^(a2, 03, ... , aj +2/^(03, 04, . . . ,aj 
-...+(-ir-^SF(aO + (-irF(0), ' 
n-l 
where S now indicates that the summation extends over all the combina- 
r 
tions of fli,. . ., a„ taken n — r at a time. The proof is just like that by 
B. Merry for Dedekind's formula. 
To give an example, let n = 2, ai = 2, 02 = 3, S = S, 4, 6, 8. Then F{ai) 
= 3, 6;/(oi)=3, /(0)=6; F(a2)=4, 6, 8; /(o2)=4, 8. Thus 
F(oi,O2)-/^(o0-F(o2)+F(0)=5-(3,6)-(4,6,8)+6=0=/(oi,O2). 
A. Berger^^'' called /i conjugate to /a if 2/i id)f2{d) = 1 f or ^• = 1 , f or A:> 1 , 
when d ranges over the divisors of k. Let g{mn)=g{m)g{n), ^(1) = 1. 
Write h{k)=Zf{d)Md)g{8), where d8 = k. Then fik)=2f.2{d)g{d)h(8). 
Dedekind's inversion formula is a special case. For, if /i(n) = l, then 
/2(n)=M(w). 
K. Zsigmondy^^ stated that if, for every positive integer r, 
?f{r;)=F{r), 
where c ranges over all combinations of powers ^r of the relatively prime 
positive integers ni, . . ., n^, while r,. denotes the greatest integer ^r/c, then 
/(r)=F(r)-2F(r„)+SF(r„„0-..., 
n n,n 
where the summation indices n, n' ,. . . range over the combinations of 
ni, . . . , rip taken 1,2,... at a time. 
R. D. von Sterneck^^ noted that, if d ranges over the divisors of n 
l^B{d) =^{n) implies that 
Taking m=l,. . ., n and solving, we get Q{n) expressed as a determinant 
of order n, whence 
^(n)=,A(^)-S,A(di)+2:V'W----+(-l)VR), 
if n = pi°'. . .p/" and d^ is derived from n by reducing p exponents by unity. 
*Here and in the statement of the theorem occur confusing misprints for k and n. 
""Nova Acta Regiae Soc. Sc. Upsaliensis, (3), 14, 1891, No. 2, 46, 104. 
"Jour, fur Math., Ill, 1893, 346. Apphed in Ch. V, Zsigmondy." 
"Monatshefte Math. Phys., 4, 1893, 53-6. 
