152 History of the Theory of Numbers. [Chap, v 
SJ.(d;m„...,m.) = [ii]W...[iL], 
where d ranges over the divisors of n, the case A; = 1 being due to Laguerre.'* 
In the latter case, take n = 1, . . . , n and add. Thus 
k=i LfcJ -^^LmJ Ini m \ m/ j 
the last equality, in which (n, h) is the g. c. d. of n, h, following from expres- 
sions for (n, h) given by Hacks^^ of Ch. XL In the present paper the above 
double equation was proved geometrically. For m = l, we get Dirichlet's^^ 
formula. The g. c. d. of three numbers is expressed in terms of them and [x]. 
The initial formulas were proved geometrically, but were recognized to 
be special cases of a more general theorem. Let 
2Md)=FM (1 = 1, ...,k), 
where d ranges over all divisors of n. Then the function 
^(n) =S/i(Xi) . . .A(X,) (1. c. m. of Xj, . . . , X* is n) 
has the property 
S^(d)=Fi(n)...n(n). 
Hence in the terminology of Bougaief (Ch. XIX) the number-theoretic 
derivative ^{n) of Fi{n) . . .Fk{n) equals the sum of the products of the 
derivatives /» of the factors Fi, the arguments ranging over all sets of k 
numbers having n as their g. c. d. 
L. Gegenbauer^^^" proved easily that, if [n, . . , t] is the g. c. d. of n, . . . , f 
2 F{[n,x^,...,x,]) = 'E F{d)jJ fj, 
where d ranges over all divisors of n, and F is any function. 
K. Zsigmondy^^^ considered any abelian (commutative) group G with 
the independent generators ^i,. . ., Qs of periods ni, . . ., n^, respectively. 
Any element g'l''' . . . gj"' of G is of period 5 if and only if 5 is the least positive 
value of X for which xhi,. . ., xhs are multiples of rii, . . ., n^, respectively. 
The number of elements of period 5 of G is thus the number of sets of posi- 
tive integers hi,. . ., hg {hi^rii,. . ., /ij^nj such that 5 is the least value of 
X for which xhi,. . . , xhs are divisible by ni, . . . , n^, respectively. The num- 
ber of sets is shown to be 
rPid;ni,...,n,)=lldjIl{l-l/q.'*), 
where 5_, is the g. c. d. of 5 and Uj] q\,...,qr are the distinct prime factors 
of 5; while U is the number of those integers nx, . . ., n^ which contain q^ 
at least as often as 5 contains it. If 5 and 5' are relatively prime, 
ypib] ni,. . ., n,)\pi8'; rii,. . ., n,)=i/'(55'; nj. . ., nj. 
"li^Sitzungsber. Akad. Wiss. Wien (Math.), 103, Ila, 1894, 115. 
"'Monatshefte Math. Phys., 7, 1896, 227-233. For his we write \p, as did Carmichael." 
I 
