History Of The Theory Of Numbers - I

300 HISTORY OF THE THEORY OF NUMBERS. [CHAP. X
Beginning with p. 425 and p. 430 there enter the two functions
/AX^r/A^
in which (A/p) is Legendre's symbol, with the value 1 or — 1.
J. W. L. Glaisher74 investigated the excess $r(ri) of the sum of the rth powers of the odd divisors of n over the sum of the rth powers of the even divisors, the sum A'r(rc) of the rth powers of those divisors of n whose complementary divisors are odd, wrote f for ft, and A' for A\, and proved
A'3(n) =nA/(n)+4A'(l)A'(n-l)+4A/(2)A/(n-2)+ . . . +4A'(n f8(n)-(2n-l)r(»)-4f(l)f(n-l)-4f(2)r(n-2)-. . . -4f(n-nA'(n) =A'(l)A'(2n-l) -A'(2)A'(2n-2) + . . . +A'(2n-l)A'(l),
A'8(n) =nAx(n) +A/(2)A/(2n-2) +A' (4)A; (2n-4) + . . . +A/(2n-2)A/(2),
+9A(4)A(n-4) + . . . +A(n-l)A(l)} (n even),
,2r-l] [3, 2r-3] [2r-l, 1]
""""""
(2r)! where
For W odd, J"(n) =A'(n) =<r(ri) and the fourth formula gives (n-l)er(n>=8{<^^ Glaisher75 proved that
(n) —6n(r(ri) +(r(n)
<r(l)cr(2n-l)+<r(3)(r(2n-3) + . - -
The latter includes the first theorem in his63 earlier paper. Glaisher76 proved for Jacobi's11 E(ri) that
(i;-16) -2<r(t>-36)+ . . . =0
and three formulas analogous to the last (pp. 125, 129). He repeated (p. 158) his74 expressions for A'3(n).
"Messenger Math., 14, 1884-5, 102-8.
™Ibid., 156-163.
"Quar. Jour. Math., 20, 1885, 109, 116, 121, 118.