History Of The Theory Of Numbers - I

CHAP. X] SUM AND NUMBER OF DlVISOES. 301
L. Gegenbauer77 considered the number Ti(fc) of the divisors ^[-\/n] of k and the number r2(&) of the remaining divisors and proved that
0(s) being 90 of the order of magnitude of s. He proved (p. 55) that the mean of the sum of the reciprocals of the square divisors of any integer is 7r4/90; that (p. 64) of the reciprocals of the odd divisors is 7r2/8; the mean (p. 65) of the cubes of the reciprocals of the odd divisors of any integer is ?r4/96, that of their fifth powers is 7r6/960. The mean (p. 68) of Jacobi's11 E(n) is
7T/4.
G. L. Dirichlet78 noted that in (7), p. 282 above, we may take e to be of lower [unstated] order of magnitude than his former \/n.
L. Gegenbauer79 considered the sum rr k>a (n) of the /cth powers of those divisors of n which are rth powers and 'are divisible by no (sr)th power except 1; also the number Qa(b) of integers ^6 which are divisible by no ath power except 1. It follows at once that, if /i,(m) =0 if m is divisible by an sth power >1, bi|t = 1 otherwise,
where the summation extends over all the divisors dr of n whose complementary divisors are rth powers, and that
(14) S rr. »..(*) =
z*l aj
From the known formula Qr(ri) =S[n/xr]/ji(#), x== 1, . . . , v, is deduced
1/=1
the right member reducing to n for fc = 0 and thus giving a result due to Bougaief. From this special result and (14) is derived
From these results he derived various expressions for the mean value of Tr,-.k,s(x) &nd of the sum rr^>s(n) of the &th powers of those divisors of n which are rth powers and are divisible by at least one (sr)th power other than 1. He obtained theorems of the type: The mean value of the number of square divisors not divisible by a biquadrate is 15/Tr2; the mean value of the excess of the number of divisors of one of the forms 4-rp,+j(j= 1,3,..., 2r — 1) over the number of the remaining odd divisors is
2
4r
"Dcnkschr. Akad. Wicn (Math.), 49, I, 1885, 24.
78G6ttinRen Nachrichtcn, 1885, 379; Werke, 2, 407; letter to Kronecker, July 23, 1858.
70Sitzungsberichte Ak. Wiss. Wicn (Math.), 91, II, 1885, 600-621.