Qfi Art. 4. —M. Kuniyeda : 



1 



as X^ X . 



The trutli of this corollary can be inferred immediatel}?- by 

 writing our equations in tlie forms 



1 1 1 



y [/(>/)] " = / " , y [/(y)} « = c " ;. n . 



19. Lemma 5. Lef. fix) and fi(x) be L-functions such 

 that 



/>o, /;>o, />/>! 



as x^O. If x = d, x = Oi ((re respectively the roots of the equations 



f{x) = K Mx) = i 



for large values of X, then 



e > d, 



for every sufficiently large value of X. 



If we notice that/ and /i are ultimately monotonie and/</t 

 for every sufficiently small value of x, then our lemma follows im- 

 mediately. 



VII Discussion of Case (C) : (y{x) > 7(l/a;). 



20. We now pass to the discussion of the behaviour of the 

 integrals* 



C(h = f l>U)r^''^''>-^-^^^dx, 



Jo X 



. s (/) = f'o(x)r:<'^^^^^^ dx, 



./ X 



as à-^:d, when / (1/a;) < ^ < (l/a;)^ It will in this case be conve- 

 nient to separate the real and imaginary parts of the integrals. 

 Thus we have to consider 



* Although the sine-integral S{\) has already been treated by Hardy, we shall discuss it 

 again, reproducing briefly his analysis, because, for the purpose of this paper, it is necessary to 

 modify his argument and to extend it to the case x'-^p^x, while the same argument applies 

 to the discussion of the cosine-integral t'().). 



