298 HlSTOEY OF THE THEOEY OF NUMBEES. [CHAP, X The first three had been found and proved purely arithmetically by Lipschitz and communicated to Hermite. Hermite proved (11) by use of series. Also, where d ranges over the divisors of n. When f(d) = 1, F(ri) becomes r(n) and the formula becomes the first one by Dirichlet.17 L. Gegenbauer72 considered the sum pkt t (n) of the &th powers of those divisors dt of n whose complementary divisors are exact tth powers, as well as Jordan's function Jk(n) [see Ch. V]. By means of the f -function, (12), he proved that Ii <r*(?n)po, 2(ri) =Sp0, zt(d)pk> t ( 3 ) > mtn d VV where d ranges over the divisors of r, and m, n over all pairs of integers for which wfl/=r; /r the latter for 2 = 1 being LiouviuVs29 seventh formula for i> the latter for < = y = 1, k = 0, being the second formula of Liouville26, while f ojr < = 1 it is the final formula by Cesaro210 of Ch. V; according as r is not or is a square; 2X(n)pfc ,(«) "P*. j,(r), 2 rCz2) =2 r2(r), S ^x(x)at(x) =2 Pk,2(r). By changing the sign of the first subscript of p3 we obtain formulas for the sum Pktt(ri)=nkp_ktt(ri) of the kilo, powers of those divisors of n which are tth powers. By taking the second subscript of /> to be unity, we get formulas for o-fc(n). There are given many formulas involving also the number fa(ri) of solutions of n^. . .na=n, and the number co(n) of ways n can be expressed as a product of two relatively prime factors. Two special cases [(107),(128)] of these are the first formula of Liouville26 and the ninth summation formula of Liouville,29 a fact not observed by Gegenbauer. He proved that, if pgn, I, B(z)=- 2 C(x)+Bn-Ap, "Sitzungsber. Ak. Wiss. Wien (Math.), 89, II, 1884, 47-73, 76-79.