Chap. V] FarEY SeRIES. 155 
K be the area of the region bounded by this curve, and N the number of 
points {x, y) within it or on its boundary such that a; is a multiple of k and 
is prime to y. Then ^ ^ 
lim— = -2Pi.fc, 
k=aa A TT 
where K increases by uniform stretching of the figure from the origin. 
In particular, consider the number A^ of irreducible fractions x/y^\ 
whose denominators are ^n. Since x^y, the area K of the triangular re- 
gion is n^/2. Hence N = {n^/2) (6/7r^) , approximately (Sylvester^^) . Again, 
the number of irreducible fractions whose numerators he between I and 
l-\-m, and denominators between V and I'+m', is Qmm'/ir^, approximately. 
There is a similar theorem in which the points are such that y is divisible 
by k', while three new constants obey conditions of relative primality to 
each other or to x, y, k, k'. 
Extensions are stated for m-dimensional space. 
E. Cahen^^^ called /^(n) the indicateur of /cth order of n. 
G. A. Miller^^° evaluated Jk{in) by noting that it is the number of 
operators of period m in the abeUan group with k independent generators 
of period m. 
G. A. Miller^^^ proved (10) and (11) by using the same abelian group. 
E. Busche^^^ indicated a proof of (10) and (12) by an extension to space 
oi k+1 dimensions of Kronecker's^^^ plane, in which every point whose 
rectangular coordinates x, y are integers is associated with the g. c. d. of x, y. 
A. P. Minin^^^ proved (14) and some results due to Gegenbauer.^"^ 
R. D. CarmichaeP^^ gave a simple proof of Zsigmondy's^^® formula for ^. 
G. Metrod^^^ stated that the number of incongruent sets of solutions 
of xy' — x'y = a (mod m) is 'EdmJ2{m/d), where d ranges over the common 
divisors of m and a. When a takes its m values, the total number of sets 
of solutions is vJ'Ay^ rt'A 
It is asked if like relations hold for Jk, k>2. 
Cordone^^ and Sanderson^^^ (of Ch. VIII) used Jordan's function in 
giving a generalization of Fermat's theorem to a double modulus. 
Farey Series. 
Flitcon^^ gave the number of irreducible fractions <1 with each 
denominator <100, stating in effect the value of Euler's (/)(n) when 
n is a product of four or fewer primes. 
"9Th6orie des nombres, 1900, p. 36; I, 1914, 396-400. 
«»Amer. Math. Monthly, 11, 1904, 129-130. 
2"Amer. Jour. Math., 27, 1905, 321-2. 
222Math. Annalen, 60, 1905, 292. 
^^'Vorlesungen iiber Zahlentheorie, 1901, I, p. 242. 
224Matem. Sbomik (Moscow Math. Soc), 27, 1910, 340-5. 
"^''Quart. Jour. Math., 44, 1913, 94-104. 
«2«L'interm6diaire des math., 20, 1913, 148. Proof, Sphinx-Oedipe, 9, 1914, 4. 
^'Ladies' Diary, 1751. Reply to Question 281, 1747-8. T. Leybourn's Math. Quest, pro- 
posed in Ladies' Diary, 1, 1817, 397-400. 
