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CHAP.V]                             ETTLER'S ^-FUNCTION.                                    139
(d— l)n; whence <j>(dri)=d<t)(ri). Finally, let p1} ..., pv be the v=(j)(n) integers <n and prime to n. Then Pi+kn (t = l, . . ., v\ & = 0, 1, . . .) give ail integers prime to n; let Ph(ri) denote the hth one of them arranged in order of magnitude. Then
Pky(n)=kn-l (fc^l),     Pto+r(n)=fcn+pr(l^rgy-l, fc^O). If fr = &j/+r, r<j>, the sum of the first h numbers prime to n is
where pi, >. . , pr are the first r integers <n and prime to n.
K. Hensel118 evaluated 0(n) by the first remark of Crelle.17
J. G. van der Corput and J. C. Kuyver119 proved that the number /(a/4) of integers ^a/4 and prime to a is N*=\dH(l — l/p) if a has a prime factor 4m +1, where p ranges over the distinct prime factors of a; but is N—2k~2 if a is a product of powers of k prune factors all of the form 4m — 1. Also /(a/6) is evaluated.
U. Scarpis120 noted that (f>(pn— 1) is divisible by n if p is a prime.
Several writers121 discussed the solution of 4>(x) = <£(?/), where x, y are powers of primes. Several122 proved that <j>(xy)> <f>(x)<j)(y) if x} y have a common factor.
J. Hammond123 proved that there are ^<t>(n) — 1 regular star n-gons.
H. Hancock124 denoted by $({, k) the number of triples (-i, k, 1), (i, &, 2), . . ., (i, k, i) whose g. c. d. is unity. Let i~iid, fc = fcid, where ii, ki are relatively prime. Then <£(£, fc)=ii$(d), <£(&, i)=
A. Fleck125 considered the function, of 7n=Hpa
Thus </>o(m) =<£(m) , 0_i (m) == m, <j>^(m) is the sum of the divisors of m.   Also 2 0*(d)=<frb_iOn),       </>A(mn)=0A(
if w, n are relatively prime.   For f (s) =2w~~*,
118Zahlentheorie, 1913, 97.
"'Wiskundige Opgaven, 11, 1912-14, 483-8.
120Periodico di Mat., 29, 1913, 138.
121Amer. Math. Monthly, 20, 1913, 227-8 (incomplete) ; 309-10.
122Math. Quest. Educat. Times, 24, 1913, 72, 106.