48 



(45) 



6, ß having the meanings defined in the theorem. 



This formula for 8{X) is nothing but tlie one obtained in 

 " 0. D. I. 2'\ In the foDowings we are going to prove that the 

 formulae (45) hold also when p<x-^(t" so long as C(A)>1/;. and 

 S(Ä)>lß as /î-^oo. 



28- For the purpose of this paper, it is necessary to compare 

 the order of magnitude of 



J^ p{a) p{d) 



as >^v->oo, when p<^x^a". 



At first, we consider the first and the last of these functions. 

 Now, since X + a'{d) = 0, we liave 



Hence, if /' < x^a"la', then 



if py- X's/o"l(r', then 



P(d) ^^ 1 



pm vi 



We know that, if o > l(l/x), then 



n' > ^cr". [H-lemma 31] 



Hence, if i' > a:, tlien 



pW >1. 



Therefore we obtain : 



