72 Art. 4. — M. Kuniyeda : 



Lemma 8. Ld f>{x) <nt(J <t{x) be the L-functions ivhich are 

 treated in Theorem VI, and let cj{x) be a re(d function, not nece>^>^arHi) 

 an L-fu)iction, but contlniioua and different ruble in the interval (0, ?) 

 save for x = 0, satisfi/ing the relation w{x) r^ p{x) in mich a leay tlad 



vi{x) = j>{x) [I + t[xy. , 



where b{x) is tdtimateh/ monotonie and tends to zero as x-^Q. If tJtere 

 exists an L-fanctioti y{x) such tliat 



then, under the conditions the same as those of Theorem VI, the same 

 asymptotic formulae (7). {8) and {49) hold respectively for the inteyrfds 

 obtained by replaciiaj w{x) for (>[x) in S{?<), G{?), 7i(/), I^i)-\ I-l?-) and 



UX). 



Proof. If t(x) be an L-function, then the lemma follows 

 immediately from Theorem \'I and its corollary. 



If e(a?) is not an L-function, still it behaves like an L-function 

 under our hypothesis and hence the truth of the lemma may l)e 

 conjectured from Theorem VI. 



Take the integral 



/(•;.) = / ' ti?(.T) cos a(x) ^^1^ dx 

 / ,r 



/"^ r \ ^^r, f \ COS Ix 7 , C^ r \ I \ f \ COS Ix. J 

 f)[x) COS fT(a-) dx + I t{x) Mx) cos a{x) dx. 

 X ./ ■ X 



Let y(x) be an L-function such that 



lix) < rix) < 1* ' ■ 



as x^O : and write 



Pix)=f>{x)r(x), e(x) = 6^, 



SO that /'</'. ^ < 1. 



since t ^ r- Then we have 



* For instance, we may take Y=[t(-k)> ■ 



