304 History of the Theory of Numbers. [Chap, x 
sth powers of the divisors of n whose complementary divisors are even, 
the excess f ',(«) of the sum of the sth powers of the divisors whose com- 
plementary'' di\'isors are odd over that when they are even, and the similar 
functions'^ A'„ f „ a,. The seven functions can be expressed in terms of any 
two: 
where the arguments are all n. Since D\{2k) =(T,(k), we may express all the 
functions in terms of a^n) and (TXn/2), provided the latter be defined to be 
zero when n is odd. Employ the abbreviation 'EfF='S,Ff for 
/(l)F(n-l)+/(2)F(n-2)+/(3)F(n-3) + . . .+/(n-l)F(l). 
This sum is evaluated when/ and F are any two of the above seven functions 
w^th s = 1 (the subscript 1 is dropped) . If 
f{n)=aa(n)+^D'in), F(n) =aV(n)+/3'Z)'(n), 
then 
2/F = aa'2(T(r+(ai8'+a'/3)S(Ti)'H-/3iS'2i)'Z)'. 
By using the first formula in each of two earUer papers,'^' '^ we get 
12'Z(ja = 5(T2{n) —6na{n) -\-a{n) , 
122D'D' = 5Ds'(n)-dnD'{n)+D'in), 
242o-D' = 2(r3(n) + (l-3n)(7(7i) + (l-6n)D'(n)+8Z)3'(n). 
Hence all 21 functions can now be expressed at once linearly in terms of 
0-3, Ds', (T and D'. The resulting expressions are tabulated; they give the 
coefficients in the products of any two of the series 2f/(n)x", where/ is any 
one of our seven functions ■vsithout subscript. 
Glaisher^^ gave the values of l^a^ai for 2 = 3, 5, 9 and ^<r^(TT, where the 
notation is that of the preceding paper. Also, if p = 7i—r, 
12 S rpa{f)a{fi) ^n^a^in) -nV(n), S rj{r)F{p) =^/F. 
r=l r=l A 
L. Gegenbauer^^ gave purely arithmetical proofs of generalizations of 
theorems obtained by Hermite'" by use of elliptic function expansions. Let 
5,(r) =2/, (7=J^5,([^]) -v&,{y), v^\yM' 
Then (p. 1059), 
The left member is knowTi to equal the sum of the ^th powers of all the 
divisors of 1, 2, . . ., n. The first sum on the right is the sum of the A-th 
powers of the divisors ^ y/n of 1, . . . , n. Hence if A^fx) is the excess of the 
"Messenp;er Math., 15, 188.5-6, p. 36. 
'•Sitzungsberichte Ak. Wien (Math.), 92, II, 1886, 1055-78. 
