Chap. V] GENERALIZATIONS OF EuLER's 0-FuNCTION. 153 
If d ranges over all divisors of the product ni . . . n^, 
Si//(5;ni, . . ., n,)=nin2. . .n,. 
d 
In case 5 divides each ni{i = \, . . ., s), 4/ becomes Jordan's Js(5). 
As a generalization (pp. 237-9) consider sets of positive integers ai, . . . , a„ 
where aj = l, 2, . . . , 7_, for j = 1, 2, . . . , s. Counting the sets not of the form 
n^ai, nf a2, . . . , n^f a, (i = l,. ..,r), 
we get the number 
n 7,-2 n \-%\ +s n f . J' ,,, ~\ - 
where (ni, n2, . . .) is the 1. c. m. of rii, n2, . . .. In particular, take 
n^P= . . . =n^f = ni (i = l,...,r), 
where ni, . . . , n^. are relatively prime in pairs, and let iV be a positive mul- 
tiple of ni, . . . , n^ such that 
Then the above expression equals 
J/(iV; mi,. . ., m,) = n T-l-S II [—1 + 2 H f-^^l - . • ., 
y=iLmyJ i j=\unjniA i,i' j=\unjnini>j 
which determines the number of sets 
tti,. . ., a, (ay = l, 2,. . ., — ;i=l,. . ., s) 
l_A/tyJ 
whose g. c. d. is divisible by no one of ni, ^2, . . . , n^. By inversion, 
S//g;^„...,m.) = n[|], 
where d ranges over the divisors of N which are products of powers of 
rii, . . . , Ur. When ni,...,ns are the distinct prime factors of N,J/{N; rrii, . . , , 
m,) becomes the function Js{N; mi, . . ., Wj) of von Sterneck.^^^ As in the 
case of the latter function, we have 
J/{N; Wi,. . ., m,)=SJi'(Xi; mi). . .//(X,; mj, 
the X's ranging over all sets whose 1. c. m. is N. 
L. Carhni^^^ proved that if a ranges over the integers for which [2n/a] 
= 2/c+l, then 
XJM = sg^ - 2s^'J, s^^ ^1'+... +m*. 
For k = l, this becomes 2(^(a) =n^ [E. Cesaro, p. 144 of this History]. 
D. N. Lehmer^^^ called Jmin) the m-fold totient of n or multiple totient 
of n of multiplicity m. He proved that, if A: = pi"'. . .pr°^ 
Jm{k'')=k^'''-''Jm{k), Jm(ky)=JM n \pr^-pr"^-'Xy, Pi) \ , 
1=1 
where \(y, pj =0 or 1 according as Pi is or is not a divisor of y. In the 
'"Periodico di Mat., 12, 1897, 137-9. 
"»Amer. Jour. Math., 22, 1900, 293-335. 
