134 History of the Theory of Numbers. [Chap, v 
RogeP^ considered the number of integers v<n such that v and n are 
not both di\'isible by the rth power of a prime. Also the number when 
each prime factor common to v and n occurs in them exactly to the rth power. 
I. T. Kaplan published at Odessa in 1897 a pamphlet in Russian on the 
distribution of the numbers relatively prime to a given number. 
M. Bauer^^ proved that, for x prime to in, kx-\-l represents 
\p{m) 4>{d^d2) 
<j) 
integers relatively prime to m and incongruent modulo m, where di is the 
g. c. d. {k, m) of k, m, and c?2= (/, m), {di, do) = 1, w^hile 
^W=0Wn{i-^} 
is the number of incongruent integers prime to m = pi^ . . . p/* which are 
represented by kx+l when k, I, x are prime to 7n. Of those integers, 
\p{m)/\l/{pi. . .pr) are di\'isible only by the special prime factors Pi, . . ., Pr 
of m. 
J. de Vries^^" proved the first formula of Dirichlet's.^^ 
C. Moreau^^ evaluated 4){n) by the method of Grunert.^^ 
E. Landau^° proved that 
„=i<^(n) 27r^ \ *= pp2_p+iy 
where e is of the order of magnitude of x~^ log x, C is Euler's constant, and 
f is Riemann's ^-function. 
P. WolfskehP^ proved by Tchebychef's theorem that the 0(n) integers 
<n and prime to n are all primes only when n = 1, 2, 3, 4, 6, 8, 12, 18, 24, 30, 
[Schatunowsky.'°] 
E. Landau^^ gave a proof, without the use of Tchebychef's theorem, by 
finding a lower limit to the number of integers k ha\dng no square factor 
>1, where t^k>Dt/S. 
E. Maillet,^^ by use of Tchebychef's theorem, proved the same result 
and the generaUzation : Given any integer r, there exist only a finite number 
of integers N such that the <t>{N) integers <A^ and relatively prime to N 
contain at most r equal or distinct prime factors. 
Alois Pichler^'* noted that (}>(x)=n has no solution if n is odd and >1; 
while (i)(x) =2" has the solutions x = 2''bc. . . (a = 0, 1, . . ., n + 1) if 
«^Sitzungsber. Bohm. GeseU., Prag, 1897; 1900, No. 30. 
"Math. Natur. Berichte aua Ungam, 15, 1897, 41-6. 
""K. Akad. Wetenschappen te Amsterdam. Verslagen, 5, 1897, 222. 
"Nouv. Ann. Math., (3), 17, 1898, 293-5. ' 
»»G6ttingen Nachrichten, 1900, 184. 
"L'interm^diaire des math., 7, 1900, 253-4; Math. Ann., 54, 1901, 503-4. 
"Archiv Math. Phys., (3), 1, 1901, 138-142. 
»»L'interm6diaire des math., 7, 1900, 254. 
"Ueber die Auflosimg der 01. <p{x) =n. . ., Jahres-Bericht Maximilians-Gymn. in Wien, 1900-1, 
3-17. 
