Chap. V] EulEr's 0-FunCTION. 139 
(d — l)n; whence 4>{dn) = d^{n) . Finally, let Pi, ..., Py be the v = (f){n) 
integers <n and prime to n. Then pi-\-kn (^ = l, . . .,v; k = 0, 1, . . .) give 
all integers prime to n; let Ph{n) denote the hth one of them arranged in 
order of magnitude. Then 
P,Xn)=kn-l (k^l), P,,+M=kn+pr {l^r^v-l, k^O). 
If h = kv-\-r, r<v, the sum of the first h numbers prime to n is 
where pi, . . . , p^ are the first r integers <n and prime to n. 
K. HenseP^^ evaluated <^(n) by the first remark of Crelle.^'^ 
J. G. van der Corput and J. C. Kuyver^^^ proved that the number 
/(a/4) of integers ^ a/4 and prime to a is N = \aJl{l — \/p) if a has a prime 
factor 4m+l, where p ranges over the distinct prime factors of a; but is 
N — 2^~^ if a is a product of powers of k prime factors all of the form 4m — 1. 
Also /(a/6) is evaluated. 
U. Scarpis^^'^ noted that 0(p" — 1) is divisible by n if p is a prime. 
Several writers^^^ discussed the solution of 4>{x)=4){y), where x, y are 
powers of primes. SeveraP^^ proved that (f){xy)>4>{x)4>{y) if x, y have a 
common factor. 
J. Hammond^^^ proved that there are ^^(n) — 1 regular star n-gons. 
H. Hancock^"^ denoted by ^{i, k) the number of triples {i, k, 1), {i, k, 2), 
. . . , {i, k, i) whose g. c. d. is unity. Let i = iid, k = kid, where ii, ki are 
relatively prime. Then ^{i, k)=ii(i>{d), $(/c, i)=ki(j}(d). 
A. Fleck^^^ considered the function, of m^Hp", 
<p,{m) = n|<^(p«) - (J)c/>(p"-^) +...+(- i)°(^y (p^-")}. 
Thus (f)o{'m) =4){'m), <^_i(m) = m, <^_2(w) is the sum of the divisors of m. Also 
S 4>k{d)=<i>k-i{m), (f>kimn)=(l)k{ni)(t)k{n), 
d:m 
if m, n are relatively prime. For f (s) =2m~*, 
^ (f)k-i{m) ^ 4>k{m) 
0.(p)=p-CI^), 0.(p^)=p^-(^l>+Ct')' ■•' 
</).(p'+'+o=p''(p-l)'^'. 
"sZahlentheorie, 1913, 97. 
"^Wiskundige Opgaven, 11, 1912-14, 483-8. 
i^'oPeriodico di Mat., 29, 1913, 138. 
i2iAmer. Math. Monthly, 20, 1913, 227-8 (incomplete); 309-10. 
li'^Math. Quest. Educat. Times, 24, 1913, 72, 106. 
^^'Ibid., 25, 1914, 69-70. 
i^^Comptes Rendus Paris, 158, 1914, 469-470. 
"^Sitzungsber. Berlin Math. Gesell., 13, 1914, 161-9. 
