CHAP. V] EULEK'S ^-FUNCTION. 131 For 6=n, r' = l, r-v 1, this becomes the former result; for r=r' = l, 5=n, it becomes S^(x) =n2, where x takes the values for which p(n/x)^ 1/2. H. W. Lloyd Tanner71 studied the group G of the totitives of n (the integers <n and prime to n), finding all its subgroups and the simple groups whose direct product is 0. E. Lucas72 proved that, in an arithmetical progression of n terms whose common difference is prune to n, there are 4>(d) terms having with n the g. c. d. n/d. If, when d ranges over the divisors of n, ^(d) = n for every integer n, then (p. 401) $(ri)*=<l>(ri), as proved by using n = l, a, a2,. . ., and n = db, cfb, . . . , where a, b, . . . are distinct primes. He gave (pp. 500-1) a proof of Perott's53 first formula by induction from N I to N, communicated to him by J. Hammond. The name "indicateur " of n is given (preface, xv) to 0(n) [Prouhet18]. C. Moreau (cf . Lucas,72 501-3) considered the C(ri) circular permutations of n objects of which a are alike, J3 alike, . . . , X alike. Thus, if a = 2, ft 4, the 0(6) = 3 distinct circular permutations are aabbbb, ababbb, abbabb. In general, . where d ranges over the divisors of the g. c. d. of a, /3, . . . , X. In the example, d 1 or 2, and the terms of the sum are 15 and 3. P. A. MacMahon73 noted that C(ri) = 1 if n = a, so that we have formula (4). His expression for the number of circular permutations of p things n at a time is quoted in Chapter III on Fermat's theorem. A. Berger73a evaluated Sfc? fc^Cfc). For a = 2 the result is Zn2/ir2+ \n log n, where X is finite for all values of n. E. Jablonski74 considered rectilinear permutations of indices a, . . ., X, with the g. c. d. D. Set a = a/D,. . .,X=X'jD, a+ . . .+\ = m = m'D. Then the number of complete rectilinear permutations of indices o/n, > X'n is The number of complete circular permutations is where d ranges over the divisors of D. If Q (D/d) is the number of rectilinear permutations of indices a, . . . , X which can be decomposed into d identical portions, SQ(Z)/d) = P(D) . Also 71Proc. London Math. Soc., 20, 1888-0, 63-83. 72Th6orie dcs nombrcs, 1891, 396-7. The first theorem was proved also by U. Concina, II Boll, di Matematica, 1913, 9. "Proc. London Math. Soc., 23, 1891-2, 305-313. 7«°Nova Acta Regiae Soc. Sc. Upsalicnsis, (3), 14, 1891, No. 2, 113. * "Comptes Rendus Paris, 114, 1892, 904-7; Jour, de Math., (4), 8, 1892, 331-349. He proved Moreau's72 formula for C(n).