Per 
853 
q are greater than the prime number p, and 
w= p* qd, 
the number of positive integers < r, congruent to /, with regard to 
the modulus 4, (Jand’ prime to each other) for which the number 
of divisors is exactly equal to w, is given by 
1 1 
—— + 0 | ——— for a=1 
and by 
awP—! (log log v)%—! art U Beul 
EO, A mel Gee) en 
log « log « 
for any arbitrary positive integral value of a, where 
se h.(a—1)! ai ee 
up! 
extended over all the positive integers u, of which the number of 
divisors is exactly equal to g and for which the congruence 
uzP—1 =] (mod. £) 
has roots z; 6 represents the number of incongruent roots of the 
congruence 
e—-l1= 1 (mod. 4). 
In order to prove this, we take the number of divisors of u 
equal to t,, and 
(Ups be for ty Ur, 
= 0 for Tij, == 10s 
We have to prove first that this function satisfies the conditions 
written in the proposition, if 
m=p—l, 
= a, 
f()=1 for m=q, 
= 0% fort de 
In order to give this demonstration, we distinguish four cases: 
1. Let e, be smaller than p—1; 
for 
up,“ 2.9 wee Pom 
the number of divisors of u 
tT, = (a, + 1) (a, + 1)... (a, + 1) 
is divisible by e, + 1, consequently by a number < p, hence 
