852 
of the functions 9, (w) and 6, (x), perfectly analogous formulae hold 
good. 
For the very reason that the function /’(w) is general it will not 
be difficult to deduce other corresponding relations; so we find 
the same results if we consider the squareless numbers composed 
of not more than » prime factors, or the integers for which the 
total number of prime factors is <7, or the integers for which the 
number of different prime factors is not greater than 7, etc. 
Application III. In an arithmetical series, the difference of 
which is / and the first term of which is prime to £, occur 
ze? v mii rd ii a 0 id 
6k log a pik p ‘ (log x)? 
numbers < wz, equal to a square multiplied by a prime number. 
That this is again a special case of our proposition appears by 
taking F'(w) equal to 1 or O, according to wu being equal or not to 
a square multiplied by a prime number. 
We have 
nik vhenee’ Usa, 
nz 
and 
Je =1, if v is a square, 
= 0, if v is not a-square, 
consequently 
EE ACME NE foe ee 
gere en e(t) u=1 & 
wam) U pik p) (u,k)= 
Riess Mer ee ik 
en (1— ;) Eik 
plik p) (vk)=1 
= ; i n(1 — =) SS je 
ie PJ oi? 
pik P 
: 1 
and we have only to substitute these values in (1), in order to find 
the relation sought. 
Application IV. If all the prime factors of the positive integer 
1) (u, k) represents the greatest common divisor of wand k, so that the number 
u in this sum assumes respectively each integral positive value prime to k. 
rn OET 
