CHAP. V] GENEEALIZATIONS or EULER'S ^-FUNCTION. 153 If d ranges over all divisors of the product nx . . . na, d In case d divides each n;(t = l, . . ., s), ty becomes Jordan's J,(5). As a generalization (pp. 237-9} consider sets of positive integers a1; . . . , a,, where a, • = 1, 2, . . . , y3- for „? = 1, 2, . . . , 5. Counting the sets not of the form n(JV *?«*..-, n^a. (i-l,...,r), we get the number where (n^ n2, . . .) is the 1. c. m. of n^ n2, . . . . In particular, take ^=...=n(?-nt. (t~l,...,r), where n1? . . . , nr are relatively prune in pairs, and let N be a positive multiple of ni, . . . , nr such that Then the above expression equals JY(#; *h,. - ., O=n ["— 1-z n f y-iLWjJ i j-iLm, which determines the number of sets rNi ai, . . . , a, (ay= 1, 2, . . .,|^— J ; j = 1, . . ., s) whose g. c. d. is divisible by no one of nly n2, . . ., n,. By inversion, J- where d ranges over the divisors of N which are products of powers of HI, ..., nr. When nlt..., na are the distinct prime factors of N, J,f(N; m1}.. ., 7tt8) becomes the function J,(N', rab..., mt) of von Sterneck.218 As in the case of the latter function, we have the X's ranging over all sets whose 1. c. m. is N. L. Carlini217 proved that if a ranges over the integers for which [2n/a] = 2/c+l, then For & = 1, this becomes 2<£(a) =n2 [E. Cesaro, p. 144 of this History]. D. N. Lehmer218 called Jm(n) the m-fold totient of n or multiple totient of n of multiplicity m. He proved that, if k = piai.. .prar, r Jm(kn) =fcm(n"1)Jm(A;), Jm(ky) = Jm(y) U ]p/noi--p/n<<w~1)X(2/> Pi) \ > where X(y,_pt-) =0 or 1 according as pt- is or is not a divisor of y. In the '"Periodico di Mat., 12, 1897, 137-9. J18Amer. Jour. Math., 22, 1900, 293-335.