Chap. V] GENERALIZATIONS OF EuLER's </)-FuNCTION. 151 
Ji+M=i:d'JMJi(^, 
which is next to the last formula of Gegenbauer's.^"^ Similarly, 
which is the case i = 1 of Gegenbauer's'^^ fifth formula in Ch. X, (Tk{n) being 
the sum of the A;th powers of the divisors of n. 
E. Weyr^^^ interpreted J2in) in connection with involutions on loci of 
genus 1. From the same standpoint, L. Gegenbauer^^^ proved (12) for 
k = 2 and noted that the value (10) of J-zin) then follows by the usual method 
of number-theoretic derivatives. 
L. Gegenbauer^^^" wrote cf)kim, n) for the number of sets of k positive 
integers ^ m whose g. c. d. is prime to n = pi°' . . . p/'' and proved a formula 
including 
[mf=Um, n)+i S {\„ . . . XT 4>k i -^ • , -^^ ) 
where (Xi, . . . , X,,) is the determinant derived from that with unity through- 
out the main diagonal and zeros elsewhere by replacing the 7th row by 
the X^th row for 7 = 1,. . ., c. The case m = n, k — l, is due to Pepin.^^ 
There is an analogous formula involving the sum of the /cth powers of the 
positive integers ^m and prime to n. 
E. Jablonski^^ used Jk{n) in connection with permutations. 
G. Arnoux^^^ proved (10) in connection with modular space. 
*J. J. Tschistiakow^^"* (or Cistiakov) treated the function /^(n). 
R. D. von Sterneck^^^ proved that 
J,{n) =SJ,(Xi)J,_,(X2) =S0(X,) . . 4{\), 
the X's ranging over all sets of integers S. n whose 1. c. m. is n. To generalize 
this, let Jk{n; mi, ... , rrik) be the number of sets of integers z'l, . . . , ik, whose 
g. c. d. is prime to n, while ij^n/mj for j = 1, . . . , k. Then 
Jk{n; Wi, . . ., m^)=SJ,(Xi; m\,. . ., 'm'r)Jk-r0^2] ^'r+i,- • •, rn'k) 
=2:Ji(Xi; mi). . . J'i(X^; m^), 
the X's ranging over all sets of integers ^n whose 1. c. m. is n, while m'l, . . . , 
m'k form any fixed permutation of mi, . . . , m^t, and J"i(n; m), designated 
<f)^"'\n) by the author, is the number of integers ^n/m which are prime 
to n. Also, 
"iSitzungsberichte Ak. Wiss. Wien (Math.), 101, Ila, 1892, 1729-1741. 
2i2Monatshefte Math. Phys., 4, 1893, 330. 
2i2aDenkschr. Ak. Wiss. Wien (Math.), 60, 1893, 25-47. 
2i'Arithm6tique graphique; espaces arith. hypermagiques, 1894, 93. 
2"Math. Soc. Moscow, 17, 1894, 530-7 (in Russian). 
"'Monatshefte Math. Phys., 5, 1894, 255-266. 
