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 N2k~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( d:m 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.