S48 
In order to bring out the significance of this proposition four 
applications are given as follows. 
Application I. Any integer > 14, resolved into prime factors, 
has a series of exponents and the question arises how many integers 
below a given limit are to be found with a given series of expo- 
nents and how many of these integers are to be met in a given 
arithmetical series, of which the first term and the difference are 
prime to each other. It is clear that the first question is a special 
case of the second. If the given series of exponents consists of 
one number and this number is equal to one, the second question 
is identical with the question how many prime numbers are to be 
found in that arithmetical series below a certain limit and the answer 
is given by formula (2); if the given series of the exponents is com- 
posed of one number m > 1, it is sought how many numbers equal 
to the mt power of a prime number occur in the arithmetical series, 
below a given limit and this is easy to calculate by means of for- 
mula (2). The question, however, becomes more intricate, as soon 
as the series of exponents consists of more than one number, 
but in that case the answer may be found by means of the propo- 
sition, for any series of exponents. Take e.g. the smallest number, 
occurring in the given series of the exponents, equal to m and 
suppose that this number occurs 7 times in this series, so that the 
given series of the exponents is equal to 
Os Oy © re Oos My My ae or Ms 
where 
dr OR ED pe 
Aj = 
Take /(u)=— 1, if the integer w, resolved into prime factors, has 
a series of exponents, equal to the given series and take /(u) — 0 in 
the other cases; take /(w) = 1, if the series of the exponents of the 
prime factors of the integer w is equal toa,,a@,,...,a@ and /(w) = 0 
in the other cases. The conditions, laid down in the proposition are 
then satisfied, viz. 
1. Fw=0, for eZ mand also for 4, Sn, Au 552; 
Des (00 fore, Sm; 
Bo). Alwijs tore SN 
for if e‚= M, 4d, =N and the given series of the exponents is (not) 
corresponding to that of wu, the series a,,a,,..., 4 is (not) corre- 
sponding to the series of the exponents of v,, so that both the funct- 
ions Flu) and fw) are in that case equal to one (nothing). 
The proposition may therefore be applied and_formula (1) gives 
the sums 
hed el Pane 
