Oscillating Dirichlet's Integrals. 59 



which appears to be quite pro])able ; and after that, a rigorous 

 proof will be given, with exception of a few special cases. 

 Let 



(50) 



S(ä) = ['Mx) e^«'(-)-^HL^ dx, 



Jo X 



^(/) = /"f>(x) ßi'M^}^^:^ dx, 



./ X 



where <t, p, (> are L-functions such that 



(51) iQ/x) <n < {Ijxy, xa' <i><n', J, < p 



as x-^0. 



Now we shall assume that, for all sufficiently large values of ^, 



m 



s{):) 



S{À) 



K, 



K being a certain positive constant, independent of X. 



We observe that this relation (52) evidently holds when o, p 

 are the functions treated in Theorems I, III and VI; namely 

 when they belong to each one of the cases 



(i) <y<l{\lx), p<a', S{ä)>IIX; 



(ii) a^AUAlx), p<o', ,S (/)>!/;.; 



(iii) (T > /(I /a-), Xy/(T"la' < p < xo'. 



In the followings it will be seen that the same relation holds also 

 in our case (51), except in a few special cases. Hence the above 

 assumption seems very likely to be admissible. 



36. With the above assumption, Ave can prove the lemma. 

 Lemma 7. If S(ä), S(?^) are the integrals of {50), then 



m < si^) 



as ^->oo. 



Proof It is convenient to separate our integrals into the 

 real and imaginary parts; the same methods appl}^ to both parts. 

 Thus we consider 



