Chap. V] EulER's 0-FuNCTION. 133 
the <j)(k) integers ^k and prime to k occurs just once among the residues 
modulo k of the integers from Ik to {l-{-l)k; taking 1 = 0, 1,. . ., p — l, we 
obtain this residue p times. Hence there are p({>{k) numbers ^pk and 
prime to k. These include <j){k) multiples of p, whence 4){pk) = {p — l)(p{k). 
For, if r is one of the above residues, then r, r+k,. . ., r-{-{p — l)k form a 
complete set of residues modulo p and hence include a single multiple of p. 
Hence 
</)(a6c...) = (a-l)(6-l)(c-l)..., 
if a, b, c, . . . are distinct primes. Next, for n = a^h^ . . . , we use the sets of 
numbers from lab. . .to (l-]-l)ab. . ., for Z = 0, 1,. . ., a°-~^b^~^ . . . — 1. 
Borel and Drach^^ noted that the period of the least residues of 0, a, 
2a,... modulo N, contains N/8 terms, if d is the g. c. d. of a, iV; conversely, 
if d is any divisor of N, there exist integers such that the period has d terms. 
Taking a = 0, 1,. . ., iV — 1, we get (4). 
H. Weber ^^ defined 0(n) to be the number of primitive nth roots of 
unity. If a is a primitive ath root of unity and /3 a primitive 6th root, and 
if a, b are relatively prime, a/3 is a primitive a6th root of unity and all of 
the latter are found in this way. Hence 0(a6) =</)(a)0(6). This is also 
proved for relatively prime divisors a, 6 of n — 1, where n is a prime, by use 
of integers a and jS belonging to the exponents a and b respectively, modulo 
n, whence a^ belongs to the exponent ab. 
K. Th. Vahlen^^ proved that, if la.^in) is the number of irreducible frac- 
tions between the limits a and /3, a>j8^0, with the denominator n, 
S/„.,(d) = [(a-^)n], ij~~^h,,{k)=i[{a-m], 
where d ranges over the divisors of n. For /3 = 0, the first was given by 
Laguerre.^^ Since /i,o(^)=<^(^), these formulas include (4) of Gauss and 
that by Dirichlet.2i 
J. J. Sylvester^ corrected his^^ first formula to read 
k^[k\ = 2Ui]H[i]t ^^U), r[n]=0(l)+. . .+c^([n]), 
and proved it. By the usual formula for reversion, 
A. P. Minin^^ solved ^(f){m)=R for m when R has certain values. The 
equation determines the number of regular star polygons of m sides. 
Fr. RogeP® gave the formula of Dirichlet.^^ 
*'Introd. thdorie des nombres, 1895, 23. 
«Lehrbuch der Algebra, I, 1895, 412, 429; ed. 2, 1898, 456, 470. 
"Zeitschrift Math. Phys., 40, 1895, 126-7. 
"Messenger Math., 27, 1897-8, 1-5; Coll. Math. Papers, 4, 738-742. 
"Report of Phys. Sec. Roy. Soc. of Friends of Nat. Sc, Anthropology, etc. (in Russian), Mos- 
cow, 9, 1897, 30-33. Cf. Hammond."^ 
«»Educat. Times, 66, 1897, 62. 
