r,,.,(n)=SM.(^|);'' 
Chap. X] SuM AND NuMBER OF DiVISORS. 301 
L. Gegenbauer^^ considered the number Ti(k) of the divisors ^[\/n] of ^ 
and the number T2ik) of the remaining divisors and proved that 
STi(/c)=5(log,n+2C)+0(V^), 
^r^ik) =|(log,n+2C-2)+0(V^), 
0(s) being ^° 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 7rV90; 
that (p. 64) of the reciprocals of the odd divisors is ttVS; the mean (p. 65) 
of the cubes of the reciprocals of the odd divisors of any integer is 7r^/96, 
that of their fifth powers is 7r^/960. The mean (p. 68) of Jacobi's^^ E{n) is 
7r/4. 
G. L. Dirichlet^^ noted that in (7), p. 282 above, we may take e to be of 
lower [unstated] order of magnitude than his former -\/n- 
L. Gegenbauer^^ considered the sum r^ k,s (n) of the kth 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 ^ b which are divisible by no ath 
power except 1. It follows at once that, if /Xg(^) =0 if m is divisible by an 
sth power >1, but =1 otherwise, 
where the summation extends over all the divisors dr of n whose com- 
plementary divisors are rth powers, and that 
(14) ST..,,(a;)= S -; hr^V.(^), v = Wn]. 
From.the known formula Qr{n) =S[n/a;'^]/z(x), x = 1, . . ., j^, is deduced 
the right member reducing to n for A; = 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{^) and of the sum t^.aj.X^) of the A;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/x^; the mean value of 
the excess of the number of divisors of one of the forms 4rjLi+y(j = l, 3, . . ., 
2r — 1) over the number of the remaining odd divisors is 
1 i cot(2LdV. 
4ri^i 4r 
"Denkschr. Akad. Wien (Math.), 49, I, 1885, 24. 
78G6ttingen Nachrichten, 1885, 379; Werke, 2, 407; letter to Kronecker, July 23, 1858. 
"Sitzungsberichte Ak. Wiss. Wien (Math.), 91, II, 1885, 600-^21. 
