849 
x a 
= F (xu) and ul (ws 
u=2 uz? 
M= 
which exactly represent the numbers sought. So we find e.g. 
The number of positive integers <x, composed of two different 
prime factors, occurring in these numbers respectively in the degree 
a and g, is for a >> B equal to 
vil / 1 
xe = I +0 ee 
log a e a ~ \ (log a)? 
ps 
The number of positive integers < w, composed of one quintuple 
and three double prime factors (these prime factors are thought 
different from each other) is equal to 
awe 1 ry 
Ke (log log x)? SX — + 0 GE . log log °) 
log « 
log x P p? 
and among these numbers — 
x 1 v 
(logloga)y? = +0 (55 . Log log ° 
log « p==l (mod. 8) p? log © 
integers are to be found, which are congruent to /, with regard to 
the modulus 8 (/=1, 3, 5 or 7) and 
4e (log log x)? = Ee +0 ee . log log «| 
log « p=t1(mod.10) pe log a 
integers, which are congruent with /, with regard to the modulus 
MOAB (or. 9): 
In the following application, viz. with the function zr, (7) defined 
there, the case will be treated that the given series of exponents 
consists of „ numbers, each equal to 1. 
Application II. We introduce the following well-known notation’) : 
zr, (a) represents the number of squareless integers < 2, composed 
of n prime factors, @, (7) the number of integers < wv, of which the 
number of different prime factors is equal ton, and 6, (@) the number 
of integers <a, for which the total number of prime factors equals 7. 
Gauss surmised in 1796 
Uh wx (log log or! 
(nl)! loge 
Hy (©) ~ 
1) See for this notation e.g. E. Lanpau, Handbuch I, pages 205, 208, 211. 
