(AB) 
based on the existence of the multiplication-theorem for the denomi- 
nator of the sign. 
4. To give a further example of the fitness of the method followed 
in the previous investigations, I will determine according to modulus 
n the sums of the tk powers of those terms of a system of rests in 
respect to it belonging to the exponent s. For this I make use of 
the relation of NasImor. 
If m belongs according to modulus ~ to the exponent s which 
must of course be a divisor of p (n), then all terms belonging to s 
of the system of rests according to modulus rz can be represented 
by the powers m* in which z assumes all real entire numbers under 
s prime to s. So the sum sz of & powers of this is 
T=sl 
ss = =) mie 
Thi 
and so according to the relation of NAsIMoFr equal to the expression 
3 s—l s 
= u(d)| = mdz 
d —— | 
in which the summation is to be extended to all divisors d of s and 
in which w (a) denotes the ordinary function of MöBrus-MERTENS. 
So we also get the congruence 
mks 
= 
8, = U (s) + = EE mkd u (d) (mod. n) (AZ 5), ..... (3) 
from which ensues immediately 
8; = MU (s) (mod. n.) 
This congruence furnishes, as we may notice by the way, for 
k>1 and Zh (k,s)=1 the relation 
= oh (z) =0 (mod. 7), 
Tv 
in which the summation according to z is to be extended to all 
divisors of &. 
