SER 
855 
hold good. It is a matter of course that such a relation does not 
„hold good for any function satisfying the condition stated in $ 1. 
It appears, however, that we have only to modify this condition 
a little to be sure that such a relation does hold good, viz. : 
If the arithmetical function /(w) of the integer u > 1 satisfies 
the conditions : 
1. for. ¢,<.m, and, also for ¢, =m, au > n, we have 
E(u) = 0; 
2 LOE ¢ys— du =n we have 
Pty. == of (tyr dy) 
where /(v,a) represents an arithmetical function of the positive integers 
v and a, and 
1 
m(m-+1) 
then there are constant values Da, for q2a21, n—12b>0 to 
be found for any positive integral value of q, satisfying the relation 
a F (u) = Oos”), where w< 
b 
] 
= 7 a Gs n—1 (log log x) am 
SUP (alias? Se DB, (oie 
poe a=1b=0 (log a)" (log «)4 
ul 
This proposition is again very general; this appears obviously by 
the observation that the functions which occur in the four appli- 
cations of this paragraph and which have been substituted for /’(w) 
also satisfy this condition, so that the formulae deduced in those 
applications are also to be intensified with this proposition. And the 
formulae obtained in application II are exactly the formulae (8). 
The proposition is elementarily, i.e. without using considerations 
belonging to the theory of functions to be deduced from the well- 
known relation 
x 
Bx pos 1 du v 
. zit h log u 1 (log vy? : ) 
paw Ei ( 
according to a reasoning somewhat similar to the one followed here 
in order to prove formula (1); it goes, however, without saying that 
the proof is not so simple. 
1) E. LANpAv. Handbuch. |. p. 468. 
