33 -^rt- 4.— M. Kuniyeda : 



and, if we write 



pÇx) = x~^^~^\ o x) = ax~^, 



tlien 



Att ,/ X 



where 



2^e-*^ + *^-'[J,(n)-^J,:«)], 



J^(n) = r ÉE1[\ j^t^.jc) + itlx)] COS [nx + o{x)} dx, 



./ X 



J^n) = f'A^[l + e,{x) + ie,(x)} sin [7ix + n{x)} clx, 

 ./ X 



ei{x) and Eal«) being real functions sucli that 



s,(x) = 0{x\ ^Ix) = Oix). 



We easil}^ see^"" that there exist certain L-functions ^^[x) and 

 y.J^x) such that 



as x^O, and also that -5-^ and — r^ have ultimately constant 

 ' ax dx 



signs. 



* From -69) iind (70), we see that 



= (l.l,.....)%os{(ä-f>.-}--l. 



\ 24 / lU2 2 / j 



Hence ei and £.j may be expressed as power series of .c, which are uniformly converuent for 

 sufficiently small values of x. Thus the first terms of these series may respectively be taken 



as Yi and fo, and ^t immediately follows that ---^- and — -^ have ultimately constant signs. 



ti.r <lx 



