CHAP. X] SUM AND NlJMBEB OF DlVISORS. 293 and the fourth of Liouville.26 Taking #=#, /=<£, we get the third formula of Liouville.26 For # = 1/z, /=0, we get For 0 = 0 or of , / = x*, we get the first two of Liouville's29 summation formulas. If TT(X) is the product of the negatives of the prime factors 9*1 of x, Further specializations of (13) and of the generalization (p. 47) led Cesaro (pp. 36-59) to various formulas of Liouville25"27 and many similar ones. It is shown (p. 64) that for f and F as in (12), (13). For /(n)=0(n), we have the result quoted under Cesaro87 in Ch. V. For f(ri) = 1 and nk, m- k> 1, If (n, j) is the g. c. d. of n, j, then (pp. 77-86) ST^T = 2S<r(d) - 1, nr(n) =Scr(n, j)> <r(n) =Sr(n, j), If in the second formula of Liouville26 we take m = 1, . . . , n and add, we get y.l LJJ y-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, 6,... for which x — cf, x — 62,... 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 is7r/(6Ar); the excess of the number of divisors 4/xH-l over the number of divisors 4ju+3 is in mean ?r/4, and that for 4^+2 and 4p is J log 2; the mean of the sum of the inverses of the odd divisors of any integer is 7r2/8; the mean is found of various functions of the divisors. The mean (p. 172) of the number of divisors of an integer which are rath powers is f(w), and hence is 7T/6 if