92 



Art. 4. — M. Kuniveda : 



We easily see* that there exist certain L-functions yi(x) and 

 yjß) such that 



as x-^0, and also that -^ and -^ have ultimately constant 



signs. 



^y hypothesis 



hence the condition 



jxl + q, 

 f < x<t' 



is satisfied, so that the integrals Ji{n), J.^n) and J{n) are convergent. 

 Thus, applying Lemma 8 of Part I. we obtain 



J\{n) <^ / rr'^ cos {nx-\-ax''^] dx, 

 ./ 



J".,(?i) <^ / rr"'' sin [w,T + «,r~'] dx 

 J 



as w^oc . Hence, we have 



/iCn) f-' ( =rr-)' (7«)'^+' « " ^+' cos 1x11^+'' + ^-), 



(2— \i i'--^ i'-i-èv ? 



and hence 



(77) 



/(n)^ l^^p^^y.} {qa) f+. >, 1+'/ exp [-(A: n^+'-i/^rr + ir)/}, 





* The exact forms of rj and t . are 



^ 2 sin è.<- / I ,r* (2 sin ^.r)" '^ / 



^.(.r) = /-—-^L __f .-" sin è '/■'■/ >2 sin è.r)'' ,5^ ( » ,_ ^tcosè? JL_w\ . 

 V 2sinir .' I .r/ (2 sin ^r)'' J 



These functions are continuous and differential ik- when x is suttioiently small, including the 

 value .T=0. And we can easily obtain the said result. 



