CHAP. V] ETJLEB'S ^-FUNCTION. 125 E. Lucas48 stated and Radicke proved that o, n) = " Jfe»2 n) is the number of integers > a, prune to a and ^ n. H. G. Cantor49 proved by use of f -functions that W-Vr2. . .ii.2*W*W . • -<K",-i) -»', summed for all distinct sets of positive integral solutions v0) . . . , vf-i of *>0 . . . pp= n, and noted that this result can be derived from the special case (4). 0. H. Mitchell80 defined the a-totient ra(k) of k^a'b". . . (where a, 6, ... are distinct primes) to be the number of integers <k which are divisible by a, but by no one of the remaining prime factors 6, c, ... of k. Similarly, the afr-totient Tab(k) of k is the number of integers <A; which are divisible by a and 6, but not by c, . . . ; etc. If fc=a%V, ra(k) =a'-V(&V), rab(k) = a'-16 If 0- contains the same primes as s, but with the same exponents as hi k, so that <r=a* if s = a, it is stated (p. 302) that C. Crone61 evaluated <t>(ri) by an argument valid only when n is a product of distinct primes PI, . . ., pfl. The number of integers <n having a factor in common with n is then 2 -- 1). i-.-Pfl-i / The sum of the second terms of each sum is Hence the number of integers <n and prime to n is n-l-A=n-2~+2— — . . . -( Pi PlP2 provided n = pl. . .pQ. [To modify the proof to make it valid for any n, we need only add to A the term and hence replace ( — l)fl by ( — l)9w/(pi- • -Pa) m n — l—A.] 48Nouv. Corresp. Math., 6, 1880, 267-9. Also Lucas,72 p. 403. "Gottingen Nachrichten, 1880, 161; Math. Ann., 16, 1880, 583-8. 60Amer. Jour. Math., 3, 1880, 294. "Tidsskrift for Mathematik, (4), 4, 1880, 158-9.