847 
§ 4. In this last paragraph we have to consider the proof and 
the significance of the formulae (1) and (3), which have been 
demonstrated in $$ 2 and 3. As to the proof, we see that relation 
(1) has been deduced from (2) in an elementary way and as has been 
observed at the beginning of the preceding paragraph, some other 
formulae, e.g. (3) may be proved by means of (1). Relation (8) may 
also be demonstrated directly, viz. without the round-about way 
along formula (1), by not starting from formula (2) but from 
the relation 
1 1 
ae == lagtog dr A OLY. me ere a a RD) 
pa P Wor wae 
pl 
This proof is analogous to the one used in order to demonstrate 
relation (1); on executing it, it will appear that in that case the 
proof is even simpler. Yet, that proof has not been given in this 
paper, because (1) lies deeper that (3), i.e. (3) is to be deduced 
from (1) and the reverse is not possible, so that it would not do 
to prove formula (3) first, as-it is not possible then to conclude to 
formula (1) and as will be seen it is principally tbis formula 
that we want. The question, however, is somewhat different for 
k=1, as (©) in that case is to be deduced’) quite elementarily 
from the identity 
a& v [z) 
= log p (B ie E dee. |= 2 log u ’) 
psx ie Bins aes 
=wlog«+ O(a) 
so that relation (3) may be proved quite elementarily for 4= 1. 
Formula (1) is also to be proved directly, i.e. without using 
(2); it is namely possible to prove (1) with propositions in the 
theory of functions in a way, analogous to the one, used to demon- 
strate formula (2); it is clear, however, that, in that case, an ele- 
mentary proof is not to be thought of and we have succeeded in 
deducing (1) from (2) by means of elementary methods. 
If in (1) and (3) u is taken equal to nothing, we have this 
Proposition. Ii the finite arithmetical function /’(w) is equal to 
nothing for e,<m and also for e, =m, a,>n, and the function /(2) 
equals nothing for e,em, F(w) being equal to f (w‚) for e, = m, au=n, 
the formulae (1) and (3) hold good, if Land / are prime to each other. 
1) E. Lanpavu. Handbuch I. p. 450. 
2) E. LANDAU. Handbuch I. p. 98—102. 
3) E‚ LanpAv, Handbuch I. p. 77, (formula 4), 
