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152 HISTORY OF THE THEORY OP NUMBERS. [CHAP, v <CIT VJk; ml9 . . ., wjk where d ranges over the divisors of n, the case k = 1 being due to Laguerre,85 In the latter case, take n = 1, . . . , n and add. Thus 2 /,(*; kasl ' fc y^m .mm m the last equality, in which (n, 6) is the g. c. d. of n, 6, following from expressions for (n, b) given by Hacks42 of Ch. XI. In the present paper the above double equation was proved geometrically. For ra = l, we get Dirichlet's21 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 where d ranges over all divisors of n. Then the function iKn) =SA(Xi) . . ./*(X*) (L c. m. of Xx, . .. . , X* is has the property Hence in the terminology of Bougaief (Ch. XIX) the number-theoretic derivative \l/(n) of Fi(n) . . .Fk(ri) 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. Gegenbauer215a proved easily that, if [n, . . , t] is the g. c. d. of n, . . . , t Z F([n, «!,..., *J) = 2 zi,..,xa*=i where d ranges over all divisors of n, and F is any function. K. Zsigmondy216 considered any abelian (commutative) group G with the independent generators 0i,. . ., ga of periods rii, . . ., na, respectively. Any element g^1 . . .gahs of G is of period d if and only if 6 is the least positive value of x for which xhi, . . . , xha are multiples of ni, . . . , ns, respectively. The number of elements of period 5 of G is thus the number of sets of positive integers hi,. . ., ha (hi^n^ . . ., /ia^ns) such that 8 is the least value of x for which x/ib . . . , xh9 are divisible by rii, . . . , na, respectively. The number of sets is shown to be y-i »-i where 6y is the g. c. d. of 5 and n,; ^i, . . ., gr are the distinct prime factors of 6; while Zt- is the number of those integers n^ . . ., na which contain #,-at least as often as d contains it. If 6 and 6' are relatively prime, S'; n1?. . ., n5)=^(66r; nj. . ., n.). 21SaSitzungsber. Akad. Wiss. Wien (Math.), 103, Ha, 1894, 115. 11(!Monatshefte Math. Phys., 7, 1896, 227-233. For his <$> we write & as did Cannichael."