CHAPTER V. 
EULER'S (^-FUNCTION, GENERALIZATIONS. FAREY SERIES. 
Number <f){n) of Integers <n and Prime to n. 
L. Euler/ in connection with his generalization of Fermat's theorem, 
investigated the number <j){n) of positive integers not exceeding n which are 
relatively prime to n, without then using a functional notation for 0(n). 
He began with the theorem that, if the n terms a, a-\-d,. . ., a+(n — l)d 
in arithmetical progression are divided by n, the remainders are 0, 1,. . ., 
n — 1 in some order, provided d is prime to n; in fact, no two of the terms 
have the same remainder. 
If p is a prime, (^(p"") =p'"~^(p — 1), since p, 2p,. . ., p^~^-p are the only 
ones of the p"* positive integers ^ p^ not prime to p"*. To prove that 
(1) <t>{AB) =4>{A)<f>{B) {A, B relatively prime), 
let 1, a, . . ., CO be the integers <A and prime to A. Then the integers 
< AB and prime to A are 
1 a . . . CO 
A + 1 A+a ... A+co 
2A + 1 2A+a ... 2A+C0 
(B-l)A+co. 
(5-l)A + l {B-l)A-\-a 
The terms in any column form an arithmetical progression whose difference 
A is prime to B, and hence include <^(J5) integers prime to B. The number 
of columns is (f>{A). Hence there are ({>{A)<f){B) positive integers <AB, 
prime to both A and B, and hence prime to AB. If p, . . . , s are distinct 
primes, the two theorems give 
(2) 0(p\ . .s«)=p^-np-l). . .s'-\s-l). 
Euler^ later used ttN to denote 4>{N) and gave a different proof of (2). 
First, let N = p'^q, where p, q are distinct primes. Among the N—1 integers 
<A^ there are p""—! multiples of q, and p'^'^q — l multiples of p, these sets 
having in common the p"~^ — 1 multiples of pq. Hence 
<^(iV)=iV-l-(p"-l)-(p'*-^g-l)+p"-i-l=p"-i(p-l)(g-l). 
A simpler proof is then given for the modified form of (2) : 
(3) iV(p-l)to-l)...(.-l), 
pq...s 
where p, q, r, . . . , s are the distinct primes dividing N. There are N/p 
multiples <N of p and hence N' = N{p — \)/p integers <A^ and prime to p. 
Of these, N' /q are divisible by q; excluding them, we have N" = N'{q — l)/q 
numbers < N and prime to both p and q. The rth part of these are said 
^Novi Comm. Ac. Petrop., 8, 1760-1, 74; Comm. Arith., 1, 274, Opera postuma, I, 492-3. 
«Acta Ac. Petrop., 4 II (or 8), 1780 (1755), 18; Comm. Arith., 2, 127-133. He took 0(1)=O. 
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