278 Dr Cramer, On the distribution of primes 



and 



r ji > ^+1 ?i ((« - log rif + 2/2) iog„ > #+1 w (log n - 1)- 



iogn>^+i ^i V(log'Ai)2 + ^2 ((Iog7i)^+r0(log?i-0' 



< 2 !^)( 1 +^) = o(A): (7), 



iogn>#+i ?i \(log?i)'- \ognJ \ty) 



. ^ r}(n) . 

 since the series z -, is convergent. 

 n log 71 ° 



In order to obtain similar evaluations for A^ and A3, we consider 



the division of (x — 2, x + 2) into sub-intervals of the length y^ 



which we have used for the definition of 2^;. Since t does not 



belong to £3;, it follows from this definition that none of the points 



of division situated in the interval {t — x'^y^, t + x^^) belong to the 



set Sx- Hence, denoting by t^ any such point of division, and by 



yo the corresponding value of y, we obtain, by Lemma 4, 



Thus, if we consider first A3, and group together the terms belonging 

 to the same sub-interval (considered as interval of variation of 

 logn), we obtain 



A3<KUKe~H—,+ . ^^.^ ,-t- , _.^ ^+... < — ...(8), 

 t \c^y- (c-t-lry^ {c + zfy^ J cty 



supposing c> 1 and denoting by K a constant independent of 

 c and t. 



Grouping together the terms of A^ in a similar way, we have 



^,<e-(^-^) 2 ,, } ,, 



/=K4) = K^) • ^'^' 



since the number of terms in each group is of the form (ye*). 



Of the remaining terms, A^ may obviously be treated in the 

 same way as A2, and A-^ in the same way as A3. Thus we see, from 

 (4) — (9), that it is possible to determine two absolute constants 

 Co and Xq such that 



A = t ^ ^^^^ > '^"^ 



t-cy.Klosn^t+cyJ'iii-Hny + y') ty 



