Dr Cramer, On the distribution of primes 



279 



for all values of ^ in (a; — 1, a; + 1) not belonging to S^^, so long as 

 c > Co and x>Xq. Hence, a fortiori, 



and 



e-(f-cy) 



> 



ty 



y 



7r-2 

 S 7;(n)>-- , 



t-cy<\o?;iiS,t+cy ''^ 



7;(n)>(7r-2)|e*-'^-" 



.(10). 



But we have 



V 

 t — cy < log« ^ f+cf/ 



t—cy<\o^7i<f-{-cy 



and 



log 2 1 



+ 2 - 



TT e" 



:,(f+cy)\ 



\ ( ]M-<-y)\ 



t+cy 

 log 2 1 



^-(t+cy)\ [ '^-{f-cy) 



e" 1 — TT I e 



log2 1 / 



= (log = (f e*) . 



Hence, if c is fixed and greater than Co, we obtain from (10), 



TT (e<+<'v) _ TT (e'-'^?') > ^ e*, 



for all sufficiently large values of x, and for all values of t in 

 (a;— 1, x+ 1) not belonging to Sa;. 



Proof of the theorem. 



Consider the set of points e*, where t passes through all the 

 points t of the set cr^jof Lemma 6. This set belongs to the interval 

 (e^~i_ e*^+i),and its measure is less than e^'^^fix- Hence if, in Lemma 

 6, we write x in the place of e^, t in the place of e* and k in the 

 place of 1 — a, this lemma will take the following form : 



Let us denote by a-^ the set of points t, belonging to the interval 



,exj, such that 



^{'+6t''^--^'- 





log^ 



