( 180 ) 
and so this sum has the value naught; the first sum, however, is 
equal to p(o). We have the result including also the previously 
mentioned special cases: 
The sum of the kt* powers of all terms belonging to the exponent 
s of a system according to modulus n in the region of the complex 
integers formed out of the biquadratic roots of unity ts congruent 
according to modulus n with the product 
where 
a= Th(k,s), t= Th (a, =) Py, (:, —). 
5. Finally I wish to make an observation referring to the ordi- 
nary function «(z). Up till now we have an analytical repre- 
sentation for but an extremely small number of numerical functions; 
such a one is, however, at least didactically of the greatest value, 
as it generally causes the student, beginning to occupy himself with 
higher arithmetic, much trouble to get at home in these functions 
determined only for integral values of the argument. For the function 
t(e) such a one given is in the eyclotomy, and yet as far as I 
know no use is made of it in the treatises and textbooks on the 
theory of numbers. It is 
Qari 
“y—=s—l s 
u{)= Zhe 3 
gl 
where the dash added to the > indicates that only the integers 
prime to s of the interval 1...s—I are to be taken. 
From this ensues directly 
Qa (ra + sy) ri 
ni Is TS 
2a) GC) = DE 
gs y= 
or for r and s prime to one another 
