History Of The Theory Of Numbers - I

148 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v
if pi, . . ., pq are the distinct prime factors of n. In fact, there are nk sets of k integers gn, while (n/p^k of these sets have the common divisor plt etc., whence
Jordan noted the corollary: if n and nf are relatively prime,
(11) Jk(nn')~Jk(n)Jk(n').
A. Blind156 defined the function (10) also for negative values of fc, proved (11), and the following generalization of (4):
(12) 2Jk(d) = nk (d ranging over the divisors of n).
W. E. Story201 employed the symbol rk(n) for Jk(ri) and called it one of the two kinds of &th totients. The second kind is the number </>*(n) of sets of k integers ^n and not all divisible by any factor of n, such that we do not distinguish between two sets differing only by a permutation of their numbers. He stated that
*
where 1, £xfc, ^fc, . . . are the coefficients of the successive descending powers of x in the expansion of (x+l)(x+2) . . .(x+k—1).
Story202 defined "the fcth totient of n to the condition K to be the number of sets of k numbers ^ n which satisfy condition K. The number of sets of k numbers :gn, all containing some common divisor of n satisfying the condition /c, but not all containing any one divisor of n satisfying the condition x is (if different permutations of k numbers count as different sets)
JL /i i Vi i >\ 6/*"-v VyV vv"*'
where 6, 6', ... are the least divisors of n satisfying condition /c, while $j, 61', . • • are the least divisors of w satisfying condition %• Here a set of least divisors is a set of divisors no one of which is a multiple of any other." E. Ces&ro57 (p. 345) stated that, if $k(x) is the number of sets of k integers ^x whose g. c. d. is prime to x, then
v^ fj\ fn+k-l\ ~,\ (J+k-l
where J" is to be replaced by J,(ri), and d ranges over the divisors of n.
J. W. L. Glaisher203 proved (12) by means of a symbolic expression for the infinite series Z/A(n)/(zn). If /x(n) is Merten's function,
Jk(n) -Spi where the summations relate to the distinct prime factors pi of n. Using
301 Johns Hopkins University Circulars, 1, 1881, 132. aw/btd., p. 151. Cf. Amer. Jour. Math.. 3, 1880, 382-7. 20sLondon, Ed. Dublin Phil. Mag., (5), 18, 1884, 531, 537-8.