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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.