138 History of the Theory of Numbers. [Chap, v 
{k<n). Thus |p;tn| = 1 or according as k and n are relatively prime or not. 
Hence 
n n 
(t>{n) = 2 \pkn\, (t>i(n) = 2 fc|pt„|, 
A=l k=l 
where <i>i{n) is the sum of the integers ^n and prime to n. 
R. Remak^^- proved Maillet's^^ theorem without using Tehebychef's. 
E. Landau^^^ proved (5), Wolfskehl's^^ theorem and Maillet's^^ generali- 
zation. 
C. Orlandi^" proved that, if x ranges over all the positive integers for 
wliich [m/x] is odd, then 20(x) = (?w/2)'^ for 7fi even (Cesaro, p. 144 of this 
History), while 20 (x) = k^ for m = 2k — l. 
A. Axer^^° considered the system (P) of all integers relatively prime to 
the product P of a finite number of given primes and obtained formulas 
and asymptotic theorems concerning the number of integers ^x of (P) 
which are prime to x. Application is made to the probability that two 
numbers ^ n of (P) are relatively prime and to the asymptotic values of the 
number (i) of positive irreducible fractions with numerator and denominator 
in (P) and ^n and {ii) of regular continued fractions representing positive 
fractions m (P) with numerator and denominator S n. 
G. A. ]Miller^^^ defined the order of a modulo m to be the least positive 
integer h such that ab=0 (mod m). If p" is the highest power of a prime 
p dividing vi, the numbers ^7n whose orders are powers of p are km/p" 
(k = l, 2,. . ., p"). Hence l^kim/p-'i {ki = \,. . ., p-'i) form a complete set of 
residues modulo 7?i=Ilpi'i. If the orders of two integers are relatively 
prime, the order of their sum is congruent modulo 77i to the product of 
their orders. But the number of integers ^m whose orders equal m is 
(t>{7n). Hence (/)(np°) =n0(p°). Since all numbers ^m whose orders 
divide d, a di\'isor of 7n, are multiples of 7n/d, there are exactly d numbers 
^m whose orders di\ide d, and (f){d) of them are of order d. Hence 
7n = 'E4>{d). 
S. Composto^^^ employed distinct primes 7n, n, r, and the v=<t>{77in) 
integers P\,...,p^ prime to Tnn and ^ mn, and proved that 
Pi, Pi+7nn, pi+2mn, . . . , p,+(r- l)wn {i = l,. . .,v) 
include all and only the numbers rpi,. . ., rp, and the numbers not exceeding 
and prime to 7nnr. Hence 4>{vmr)=4>{m7i)-{r — \). A like theorem is 
proved for two primes and stated for any number of primes. [The proof is 
essentially Euler's^ proof of (1) for the case in which J5 is a prime not divid- 
ing a product A of distinct primes.] Next, if d is a prime factor of 7i, the 
integers not exceeding and prime to dn are the numbers ^ n and prime to n, 
together with the integers obtained by adding to each of them n, 2n, . . . , 
"2.\rchiv Math. Phys., (3), 15, 1909, 186-193. 
i^Handbuch. . .VerteUung der Primzahlen, I, 1909, 67-9, 229-234. 
"♦Periodico di Mat., 24, 1909, 17&-8. 
"'Monatshefte Math. Phys., 22, 1911, 3-25. 
"«.\mer. Math. Monthly, 18, 1911, 204-9. 
"'II Boll, di Matematica Gior. Sc.-Didat., 11, 1912, 12-33. 
