Chap. X] SuM AND NuMBER OF DiVISORS. 293 
and the fourth of Liouville.^^ Taking g = x, f=(t>, we get the third for- 
mula of Liouville.^^ For g=l/x,f=(j), we get 
S#(d)(7 0) 
^ =Sdl 
For g=(f) or x'',f=x% we get the first two of Liouville's^^ summation formulas. 
If ir(x) is the product of the negatives of the prime factors 5^ 1 of a:, 
Sx(d)*(d)<70)i = T(»), ST(d)<#,(d)J3 = ^,2#(d). 
Further specializations of (13) and of the generalization (p. 47) 
2G(d)/(^) =i:F(d)g(^fj, F{x)^l:^|^{^)f(^^y G(a;) ^2,^(5)^ (^), 
led Cesaro (pp. 36-59) to various formulas of Liouville^^"^^ and many- 
similar ones. It is shown (p. 64) that 
n=l fl' n=l "• 
for f and F as in (12), (13). For /(n) =(^(n), we have the result quoted 
under Cesaro^^ in Ch. V. For/(n) = 1 and n'', m — k>l, 
^—^=r{m), S-— - = ^(m)f(m-A;). 
n=l '«' n '«' 
If (n, j) is the g. c. d. of n, j, then (pp. 77-86) 
.ST^-jr = 2S(7(d)-l, nT(n)=i:a{n,j), <j{n)=^T{n, 3), 
S (Tk{n, j)=n(Tk-i(n), ^j(T{n, j) =-^lnT{n)-\-a{n)}. 
y=i ^ 
If in the second formula of Liouville^^ we take m = l,. . .,n and add, we get 
s0(i)rr?i=s<T(i). 
Similarly (pp. 97-112) we may derive a relation in [x] from any given relation 
involving all the divisors of x, or any set of numbers defined by x, such as 
the numbers a, h,. . . for which x — a^, x — W,. . . are all squares. Formula 
(7) is proved (pp. 124-8). It is shown (pp. 135-143) that the mean of the 
sum of the inverses of divisors of n which are multiples of k is 7rV(6A;^) ; the 
excess of the number of divisors 4)U+1 over the number of divisors 4^i+3 
is in mean 7r/4, and that for 4/x+2 and 4ju is ^ log 2; the mean of the sum of 
the inverses of the odd divisors of any integer is ttYS ; the mean is found of 
various functions of the divisors. The mean (p. 172) of the number of 
divisors of an integer which are mth powers is f(^)j and hence is 7rV6 if 
