(76) 



plb 



It is worth noticing that this limit is independent of h. 



II. When (/>1, then /; being eqnal to dB and h to c/^ the 



numbers mh-\-h^.,c without a quadratic factor are identical with 



the Yïwmhevè d {mB -\- H) '^x, where mB -{- H is without quadratic 



factor and prime to d. Now the numbers mB -\- H (comp. § 2) 



(B4)(p{d) 

 prime to d break up into q =^ - — 77^—- arithmetic progressions with 



(f{B4) 



the difference h and the first term lij, (P. = 1, 2, ... , q), Id, being 

 prime to b; the number of (he integers without quadratic factors <[ — 



in each of these progressions amounts according to (20) to 



/ .v\ X 6 1 

 «'•">UJ = T? ' .v + 0(,/.); 



h 



n('-,^) 



Pib 

 hence 



^«r-^ ^ A^ 6 (B,d)(f{d) ^, ^ 



ii»«yi^ = l ^-MlS^ >0. . . (22) 



^ 9. We now denote by Rb,h{x) resp. by Si,,h{^) the number of 

 integers mb + /^ <C -^' containing an even. resp. an odd number of 

 distinct prime factors and we put 



P,,,(..) = ^'ft(^), 



k=\ 



where 2 ' denotes that k passes through every number mb + h ^x. 

 Then it is evident that 



z 

 k=l 



and 



