Chap. V] EulEr's 0-FunCTION. 125 
cf>{N)p-(t>iN')prcf>{N")p.. . . =<f>(NN'Nr . .)p'P'P". . .. 
E. Lucas^^ stated and Radicke proved that 
a=l fc=2 0=1 k=2 
if ^(a, n) is the number of integers > a, prime to a and ^ n. 
H. G. Cantor^^ proved by use of ^-functions that 
Svo^-Vr". . .vU24>M4>{vi) . . .0(^-i) ^n", 
summed for all distinct sets of positive integral solutions Vq,..., v^^i of 
Vq. . .Vp=n, and noted that this result can be derived from the special case (4). 
0. H. MitchelP° defined the a-totient Taik) of k^a'b''. . . (where a,h,. .. 
are distinct primes) to be the number of integers <k which are divisible 
by a, but by no one of the remaining prime factors 6, c, ... of k. Similarly, 
the a6-totient Tabik) of k is the number of integers <k which are divisible 
by a and h, but not by c, . . . ; etc. If /c = a'6V, 
tM =a'-V(&V), Ta,{k)=a'-'h--'<t>{c^), TaUk)=a'-'b^-'c'-\ 
<f>ik) +2t,(A;) +2ra,(/c) +Tadk) = k. 
3 3 
If a contains the same primes as s, but with the same exponents as in k, so 
that o- = a' if s = a, it is stated (p. 302) that 
■w=i*a- 
C. Crone" evaluated (^(n) by an argument valid only when n is a product 
of distinct primes Pi,...,Pq. The number of integers <n having a factor 
in common with n is then 
A=2(ii-l)-s(^-l) + ...+(-l).2(^^ 1). 
The sum of the second terms of each sum is 
-(0+a)- -(-^)'G^)=-i-(-i)"- 
Hence the number of integers <n and prime to n is 
n-\-A=n-l^—^^— — . . . -(-1)«S +(-!)« 
Pi V\V2 Pl-Pg-l 
provided n = pi. . .pg. [To modify the proof to make it vahd for any n, 
we need only add to A the term 
and hence replace (-1)*' by (-l)%/(pi. . .p^) in n-l-A.] 
*8Nouv. Corresp. Math., 6, 1880, 267-9. Also Lucas, '^ p. 403. 
"Gottingen Nachrichten, 1880, 161; Math. Ann., 16, 1880, 583-8. 
"Amer. Jour. Math., 3, 1880, 294. 
"Tidsskrift for Mathematik, (4), 4, 1880, 158-9. 
