Dr Cramer, On the distribution of primes 277 



the relation just stated for /x^, and then proving that we may 

 choose c so that 



7r(e< + «2')-7r(e«-«?')>f e* (3) 



V 



in the part of the remaining set which belongs to {x — 1, x + \). 

 By Lemma 5, we are able to divide {x — 2, x-\-2) into sub- 

 intervals of the length 



y-' x + 2^ 

 in such a way that not more than 



2/2 



of the points of division belong to the set S^ of Lemma 2. 



Supposing this to have been done, we exclude from the interval 

 {x — 2, x + 2) the set 2^; defined in the following way: we take 

 first the whole set S^ and then, denoting by to any of the just 

 mentioned points of division belonging to S^, the interval 

 (ta — 2x'y.2, to + 2xr'y.2). The measure of S-g is thus less than 



M ( -^~" 



M^ + 2x^y^.^=0\afe ^ ' 



In order to prove the lemma, we now have to show that c may 

 be so chosen that (3) is valid for any t\n{x — '\,x + 1) not belonging 

 to Sa;. It is to be noticed that the definition of "^^ in no way 

 involves c. 



Since t does not belong to 2*, it does not belong to S^. Thus 

 we have by Lemma 3 



i vJ±__^'rLzl (4) 



^n{{t-\ognf-^f) ty ^ ' 



for all sufficiently large values of x. We put 



00 



S=2+ S + t + 2 



1 log«</'-l t-\<\osn<t-x-yi t-x-y2<\o^n<t-ei/2 t-cyi<\ogn<t+cy2 



+ 2 + 2+2 



t+cy2<\ogn<t-i-x-yi f+A-'-i/o < log » < <+l ^+l<logM 



= A, + A.,+ ...Aj (5). 



Then we have 



A< 2 '^)< S i=o(i) (6), 



