843 
z21=M7 (a+), 
div piv 
= 1 
Me pio pte 
Re 
The quantity — — is limited (u taken fixed!) for an invariable 
Pp 
value of p and variable a—=1,2,..., since it is equal to nothing 
1 
for a—o; for any p22” and. any value a>1 it is even <1, 
existing in that case the inequalities 
a-+1 #5 a-+1 = 
I; 
per = Oa 
1 
Therefore, if v contains one or more prime factors > 2”, we have 
a+l 
1 
and as there exist only a finite number of prime numbers p< 2” 
= 
lass bak : : : 
“is limited, i.e. smaller than a number independent of v. 
vr 
Lemma. Let 7, and », be two arbitrary positive integers, whose 
sum n, +- 2, is equal to 2 and suppose /’ to be an arbitrary function 
with the parameters 7 and m; three functions /,, /’,, and /, may 
be found then in such a way that the parameters of #, are equal 
to m and n,, of HF, equal to m and n, and of F, equal to m and 
n-—1 with the relation f 
/ 
n/n! 
Pw) = EF (OF, (G) +o. fires Fee) 
(mn, Fn)! din 
Proof. Introduce the functions /’,, /',, and /’, by means of the 
1 2 3 v 
following relations: 
D= (0) OP En == My = Mis 
== 0 in the other cases, 
DE for e‚ = M, Au =N, Vu == 1, 
= in the other cases, 
Liu) =v, for ¢ > m,and also for Ee, =m, a, <n—1, 
ze) in the other cases, i.e. for e, < m and also for 
Eu =™M, Ay > n — 1. 
54* 
