150 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v where t ranges over the integers ^n which are prime to n, while p^ p2,... denote the distinct prime factors of n. If /(Z)s=l, then $(n, d)=n/d and (13) becomes ~~n Pl PlP2 ' " \ Pi/ V P2/ " " Next, take /(O = a0+a1«+o2«2+.... Using hyperbolic functions, )=4 coth (2/2)=-+~-=L+..., provided zr be replaced by n%(ri)J_r(n), where /i(n) =/'(n) -0!, /a(w) =/"(n) -2a2,.. ., Hence, since ^(w) =tf>(n), In particular, for/(0 = ^, we get </>A(n). In Prouhet's18 first formula, 6 may be replaced by the g. c. d. A0i 6 of a and b. The generalization Jk(ab) =Jk(a>)Jk(ty-jr~T~^ is proved. From (12) we get by addition* (14) Taking n=l, 2,..., n, we obtain equations whose solution gives Jk(n) expressed as a determinant of order n in which the elements of the last column are 1, l+2fe, 14~2*+3*, . . ., while for s<n the 5th column consists of s — 1 zeros followed by s units, then s twos, etc. For s>0, the element in the (s+l)th row and rth column in Glaisher's203 first determinant is 1 or 0 according as r/s is integral or fractional. J. Vdlyi207 used J%(ri) -r-0(n) in his enumeration of the n-fold perspective polygons of n sides inscribed in a cubic curve. H. Weber208 proved (10) for k = 2. L. Carlini209 gave without references (10), (11), (12), with 0(*) for Jn(k). E. Cekro210 noted that (12) implies (10). For, if S/(d) =F(n), we have by inversion (Ch. XIX), f(ri) =^(d)F(n/d). The case / = Jl gives The latter is a case of G(n) =2g(d) and hence, with (12) and *This work, Mess. Math., 20, 1890-1, p. 161, for fc = l, is really due to Dirichlet.21 Formula (14) is the case p = l of Gegenbauer's, p. 217. 207Math. Nat. Berichte aus Ungarn, 9, 1890, 148; 10, 1891, 171. 208Elliptische Functionen, 1891, 225; ed. 2, 1908 (Algebra III), 215. lo»Periodico di Mat., 6, 1891, 119-122. ™Ibid., 7, 1892, 1-6.