148 History of the Theory of Numbers. [Chap, v 
if Pi, . . ., Pg are the distinct prime factors of n. In fact, there are n*" sets 
of k integers ^n, while {n/piY of these sets have the common divisor pi, 
etc., whence 
k 
+ .... 
«"'"•-©•-©■- -(^y 
Jordan noted the corollary: if n and n' are relatively prime, 
(11) J,{nn')=J,{n)J,{n'). 
A. Blind^^^ defined the function (10) also for negative values of k, proved 
(11), and the following generalization of (4): 
(12) 2Ji.(d) =n''' (d ranging over the di\'isors of 7i). 
W. E. Story^°^ employed the s>Tnbol r'^in) for Jk{n) and called it one 
of the two kinds of kth totients. The second kind is the number </)*(n) of 
sets of k integers ^?? and not all di\isible 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 
<t>\n) =|-,ir*(n)+f,V-nn)+t2V-2(n)+ . • • +<tiT(n)[ , 
where 1, fi*, W,. . . are the coefficients of the successive descending powers 
of X in the expansion of (x+l)(x+2). . .{x-\-k — \). 
Story-°- defined "the kih. totient of n to the condition k to be the num- 
ber of sets of k numbers ^ n which satisfy condition k. The number of sets 
of k numbers ^n, all containing some common di\'isor of n satisfying the 
condition k, but not all containing any one di\'isor of n satisfying the con- 
dition X is (if different permutations of k numbers count as different sets) 
'^\H''^-y~b,')y~b,'>) 
where 5, 5', . • • are the least divisors of n satisfj-ing condition /c, while 
5i, 5/, . . . are the least di\'isors of n satisfying condition x- Here a set of 
least divisors is a set of divisors no one of which is a multiple of any other." 
E. Ces^ro" (p. 345) stated that, if $;.(a:) is the number of sets of k integers 
^x whose g. c. d. is prime to x, then 
where J* is to be replaced by J/n), and d ranges over the di\'isors of n. 
J. W. L. Glaisher-^^ proved (12) by means of a symbolic expression for 
the infinite series 2/t(n)/(a:''). If ^t(n) is Merten's function, 
JM -2p,V,(^) +2pi V«/.(^^) - • • =M(n), 
where the summations relate to the distinct prime factors p, of n. Using 
"* Johns Hopkins University Circulars, 1, 1881, 132. 
^^Ihid., p. 151. Cf. Amer. Jour. Math.. 3, 1880, 382-7. 
»<»London, Ed. Dublin Phil. Mag., (5), 18, 1884, 531, 537-8. 
