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halls, notwithstanding the simplified representation given to them 
by Max Manpr in his work ,Ueber die Verallgemeinerung eines 
Gaussischen Algorithmus” appearing in the 1st vol. of the „Monats- 
hefte für Mathematik und Physik” (1890). As far as I know no 
such method was tried for the remaining restcharacters. 
Hence the communication will not be quite worthless of the 
highly simple manner which I generally make use of in my lectures 
on the introduction of the above named generalised symbol by means 
of the explained process, the more so, as it is very fit for the 
transparent treatment of other arithmetic problems, as I wish to 
prove in the following lines in one example at least. In my 
developments I shall confine myself to the complex integers formed 
out of the biquadratie roots of unity which I shall name shortly 
integers; however, I draw attention to the fact, that this method can 
be used for all kinds of numbers for which holds good the Euclidian 
algorithm of determinating the greatest common divisor. 
1. As a basis for my developments I make use of the following 
formula 
=(n)} sn sim ( ar 
Ss {=> eed Ld). zin ON Me CLM 
>; Ll 
T=| mn; c oi 
for the sum of the values, which are obtained by a given function 
f(x). when its argument assumes all numbers, prime to a complex 
integer, of a complete system of restcharacters (except the naught) 
according to the modulus m, in which formula the summation ac- 
cording to d is to be extended to all the divisors of lying 
in the region {m} and the numerical function u () has the value + 1 
when z is a real or a complex unity of the biquadratic cyclotomic 
body or is compounded of an even number of prime numbers (one 
or two members) all of which differing mutually, and the value — 1 
when the argument is a product of an uneven number of different 
prime factors, and finally the value naught when it is divisible by 
the square of a prime number. So this function is the function of 
Moésius-Mertens for the region of the complex integers formed 
out of the biquadratie roots of unity, of which I repeatedly made 
use in former treatises (see as an example ,Zur Theorie der aus den 
vierten Einheitswurzeln gebildeten complexen Zahlen”, Denkschriften. 
der mathematisch-naturwissenschaftlichen Classe der kais. Akademie 
