118 History of the Theory of Numbers. [Chap, v 
where Pi,. . ., p^ are distinct primes. For a prime p, not dividing y, we 
have (f){py) = {p- l)<t>iy) . Take y = Pi, p = P2', then 
<^(PiP2) = (Pi-l)(P2-l)- 
Next, take y = PiP2, P = P3} and use also the last result; thus 
<A(PlP2P3) = (Pl-l)(P2-l)(P3-l), 
and similarly for <t){z). When f ranges over the integers < z and prime to z, 
the numbers vz-\-^ {v = 0,l,. . .,n — l) give without repetition all the integers 
<Z and prime to Z. Hence (f>{Z)=n<i){z), which leads to (2). [Cf. Guil- 
min,2^ Steggall.'^^] 
The proofs of (4) by Gauss^ and Catalan^ ^ are reproduced without refer- 
ences (pp. 87-90). A third proof is given. Set N = a''h^c' . . ., where a, b, 
c, . . . are distinct primes. Consider any divisor e = b^"^' . . . of N such that e 
is not divisible by a. Then 
<t>i€a'')=a''-\a-l)(t>{e). 
Sum for /b = 0, 1, . . . , a; we get a°0(e). When k ranges over its values and 
/3i over the values 0, 1,. . ., j3, and 71 over the values 0, 1,. . ., 7, etc., ea* 
ranges over all the divisors d of iV. Hence 20 (d) =a''S0(e). Similarly, if 
Ci range over the divisors not divisible by a or b, 
S</)(€)=6^(^(ei),. . ., S<^(d)=a»6^ . . =N. 
E. Prouhet^^ proposed the name indicator and symbol i{N) for 0(iV). 
He gave Gauss' proof of (1) and Catalan's proof of (4). If 5 is the product 
of the distinct prime factors common to a and b, 
<j>{ab) =(i>{a)(}>ib)8/(f){8). 
As a generalization, let 5^ be the product of the distinct primes common to 
i of the numbers Oi, . . . , a„; then 
2 § 2 2 n-l 
</)(ai. . .a„) =<j>{ai) . . .<f){an) ^ ^ 
Friderico Arndt^^ proved (1) by showing that, if x ranges over the 
integers <A and prime to A, while y ranges over the integers <B and prime 
to B, then Ay-{-Bx gives only incongruent residues modulo AB, each prime 
to AB, and they include every integer <AB and prime to AB. [Crelle's^^ 
first theorem for n = 2.] 
V. A. Lebesgue^° used Euler's^ argument to show that there are 
Nip-l){q-l)...{k-l) 
p-q. . .k 
integers < iV and prime top,q,...,k, the latter being certain prime divisors 
of A'' [Legendre,^ Minding^]. 
"Nouv. Ann. Math., 4, 1845, 75-80. 
"Jour, fur Math., 31, 1846, 246-8. 
"Nouv. Ann. Math., 8, 1849, 347. 
