( 306 ) 



and it will be found that this series has a definite snm, whatever 

 jiiay Ite the integers h and // [li<^l)). 



Firstly consider the case // = and suppose h to be prime. As 

 it is necessary to start with (inite series we write, (j being any 

 positive number, 



vib < (J inb-\-h <fi 



Then, since fi(/////) is either or — n{iii) according as ui is 

 divisible l»y h or not, we have 



h=zb—\ (/ q fj 



1 ,r^ „.T 







or 





and in the samo wav 



6,0 f, 1;0 ' /, 6.U' 



(A) 



rr '-'- 



6» 



r„6=' 



b,0 I, 1,U ' /, /-,( 



Supposing 7 to lie between h' and A^+' we niter from these 

 equations 



T-' =_vlr^ (S) 



Now it follo^vs from the identity 



m < ff 





that 



m < fi 

 »i=l 



H{m) 



is always finite and less than 2 ; hence equation (/>) gives 



2 



^.7 



6,0 



< 



6—1 



