851 
= F (u) = on (2). 
u—? 
In each of these three cases the function /(w) satisfies the conditions 
stated, if in them 
m=1 and consequently 6=1, 
fay=4 | 
and FAS 0; for dat 
so that a possesses the value — ——, and we conclude, that the 
h.(n—1)! 
relations (1) and (8) are modified to the formulae 
> E (u) = Belo: O | eogioga) 
u=? h(n—1)!log x | log wv 
u=l 
and 
tr E(u log log x)" 
= Ze he oh 4- O (log log x). 
u=2 U hin! 
For £=1 and consequently 4—41 the first of these relations 
passes into the formulae written down for zr (w), 6, (7) and en (7), 
and the second relation produces an asymptotical expression, not of 
the number but of the sum of the reciprocals of the integers 
considered, e.g. the sum of the reciprocals of all squareless numbers 
composed of n prime factors <a, is equal to j 
loql n 
(log 109 40 (log loge) "1 
n. 
and the same holds good for the numbers that are mentioned in 
the definition of v‚ (7) or on («). These formulae concerning the sum 
of the reciprocals being special cases of formula (3), where / has 
the value 1, may be proved by means of a merely elementary 
reasoning, as has been observed at the beginning of this paragraph. 
By giving an arbitrary value to # in the formula, however, we 
find that the number of squareless numbers Sw, composed of 7 
prime factors and congruent to /, with regard to the modulus &, is 
equal to 
a(log log ayr~* je w{log log x)"—* 
h.(n—l)lloga | 
and that the sum of the reciprocals of these numbers is equal to 
log log x)" 
\ ig tog OF + O(log log x)—!, 
hint! ne 
while again for the integers that are mentioned with the definition 
