(179 ) 
If o is the greatest common divisor of & and s, so that 
k= k,o, s= 38,06, Th(k,8,)=1, 
then all terms on the right side of the congruence (3) in which d 
is not a multiple of s; are divisible by x; hence we get 
— oO 
= = u (s, 0) 5 (mod. »), 
where the summation according to 0 has to include all divisors of o, 
so that specially for 
we have 
ZU (=) p (5) (mod. x), 
where by p (6) we generally denote the ordinary function p of 
GAUSS. 
If however sj and o have the greatest common multiple ¢, so that 
also 
kie to Th (a, aj) = 1 
and if farther 
eae yy Oy = TL. Oo, LR (Ty; Og) 1, 
we have the congruence 
t 
se SE (92 we (te + ZS u (% di) eo (nod. n), 
dy O» 0; 
3 
in which the summation according to 0, is to be extended to all 
divisors of 5, the summation according to 0), however, to all the 
remaining divisors of o. As each one of the numbers 0, has a divisor 
(except 1) in common with s,, each argument of the functions ge (z) 
appearing in the second sum is divisible by a square (except 1) 
12* 
