154 History of the Theory of Numbers. [Chap, v 
second formula the product equals the similar function of y' if y and y' are 
congruent modulo pip^ ■ ■ Pr- Consider the function 
U/k] 
1=1 
where m, n, k are positive integers and x is a positive number. Then if 
S{x, k) denotes l*+2''-f . . . H-[x]*, it is proved that 
which for m = n = 1 becomes Sylvester's^^ formula. By inversion, 
where ai(i) is Merten's function. For k as above and k' = k/pr''^, 
^, 
.(X, n, A;)=p,-(v-i)|(p^-i)ci,^(^^, n, A:')+$^(^, n, p,k') | 
= p,'"(«r-i)(p^_l)Sp^-^-i),^^/'^, n, A;'), 
where I is the least value of j for which [x/p^"'""^-'] = 0. Hence $^(x, n, k) 
can be expressed in terms of functions $,„(?/, n, 1). True relations are 
derived from the last four equations by replacing n by 1 — n and ^m{Xy 1 — n, 
A:) by 
\xlk\ 
n^{x,n,k)=i:j^{ik)'(ik)-'''". 
1=1 
Proof is given of the asymptotic formula 
„»n7i+l p 
^'"(^' ^' ^)=:;;;:;rxT 7r^+^' hl^^x-- log x, 
wn + l D^+i 
where A is finite and independent of x, ??2, n, while 
« 1 *■ Pi — I 
Dm+i = 2 — q:Y> P^. fc = n a^_w „+i 7TJ P„,. 1 = 1. 
j = U i=lPi \Pi —i-j 
For m = n = fc = l, this result becomes that of Mertens^* (and Dirichlet'^). 
The asymptotic expressions found for ^^i^, n, k) are different for the cases 
n = l, n = 2, n>2. 
A set of m integers (not necessarily positive) having no common divisor 
> 1 is said to define a totient point. Let one coordinate, as x^, have a 
fixed integral value 5^0, while Xi,. . ., x^-i take integral values such that 
[xi/x^],. . ., [X;„_i/X;„] have prescribed values; we obtain a compartment in 
space of m dimensions which contains /m-i(^m) totient points. For 
example, if m = 3, X3 = 6, and the two prescribed values are zero, there are 
24 totient points (xi, X2, 6) for which 0^Xi<6, 0^X2<6, while Xi and X2 
have no common divisor dividing 6. For Xi = l or 5, Xo has 6 values; for 
Xi = 2 or 4, X2=l, 3 or 5; for Xi = 3, X2 = l, 2, 5; for Xi = 0, X2 = 0, 1, 5. 
Given a closed curve r=f{d), decomposable into a finite number of seg- 
ments for each of which f{d) is a single- valued, continuous function. Let 
I 
