298 History of the Theory of Numbers. [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, 
i F{a)=i r^l/(a), F(n)^2/(d), 
a = l a = l LuJ 
where d ranges over the di\'isors of n. When f{d) = I, F{n) becomes T{n) 
and the formula becomes the first one by Dirichlet.^^ 
L. Gegenbauer"'- considered the sum p^. , (n) of the A:th powers of those 
divisors d, of n whose complementary divisors are exact ^th powers, as well 
as Jordan's function Jk{n) [see Ch. V]. By means of the f -function, (12), he 
proved that 
2 (Tk{m)po, 2(n) =2po. 2t{d)pk, t K j ' 
where d ranges over the divisors of r, and 7n, n over all pairs of integers for 
which mv} = r', 
2 Jtk{n)p,, t{m) = rV^jt. <(r) , 2 <T,_k{m)T{n)m'' ='Epk, t{d)p,, t \j) ' 
the latter for t = l being Liouville's^® seventh formula ioT v = 0; 
2dV,(^) =2dV..(0, 2/.(d)dV.(0 =P.+.<(r), 
the latter f or ^ = t- = 1 , A- = 0, being the second formula of Liouville"^, while for 
< = 1 it is the final formula by Cesaro^^° of Ch. V; 
2X(d)dV.,2.(0 =2X(d)p,.,((i)p*.,^0 =0 or P2k,t{\^), 
according as r is not or is a square; 
2X(n)p,. M =p,,2t{r), 2X(d)T(d2) =Hr)T{r), 
^r\d)J,Q =r^^, 2dV(d2)(7,(0 =2dV(d), 
2 \l\r{x') =2 At), 2 r^1x(x)(7,(x) =2 p^r). 
2=iLa;J r=l i=lL3;J r=l 
By changing the sign of the first subscript of p, we obtain formulas for the 
sum Pk,i{n)=n''p_i,j{7i) of the A:th powers of those divisors of 7i which are 
tth powers. By taking the second subscript of p to be unity, we get formulas 
for (Tkin). There are given many formulas invohnng also the number 
fain) of solutions of nin2. . .71^ = 71, and the number co(«) 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 Liou\ille-^ and the ninth 
summation formula of Liouville,^^ a fact not observed by Gegenbauer. He 
proved that, if p^n, 
2 B{x) = - 2 Cix)-\-Bn-Ap, 
i=p+l x = A + l 
"Sitzungsber. Ak. Wiss. Wien (Math.), 89, II, 1884, 47-73, 76-79. 
