829 
ers LOES SEO) 
Del A il ih v=V r+ ee 
am ym pm 
x v 1 
8 10, of 
Ol 8 log v 
ym 
Identity. If w, and w, represent two arbitrary arithmetical func- 
tions, the sum 
= w,(d,) w,(d,), 
did, Sa 
extended over all the positive integers d, and d,, of which the 
product is not greater than 2, is equal to 
fie SE Ue iy 
where 
es | 
? ENEN 
Ee nes 
x 
Ws dy 
=» ¥,(d,) RNC 
dit iit 
; Id Al We 
Ws DN YW, (d,) 
qi 
Ee 
and ijn = raf gs (d,). 
Proof. A term w, A pe ), occurring in the sum in question, 
appears in the formula 7, + 7’, — 1,7, | 
ford, SW ER Pe: exactly 1 +1—1=1 times, 
for de SV > Va exactly 1+ 0—O=1 times, 
ford > 1a de exactly 0 +1 —O= 1 times. 
Lemma. If we take 7 =1, the sum 
= Je), 
pro<Sa 
p™v = 
extended over all the positive integers v and all the prime numbers 
p, for which the relations 
p™vSex and pmy=l 
exist, is equal to 
