280 Dr Cramer, On the distribution of primes 



ivhere c is a positive constant and < ^ ^ |. Then it is possible to 

 give such a value to c that the measure fij of a^' satisfies the relation 



Suppose now that c has been properly fixed. Then it is clear 

 that, if aj is sufficiently large, and if jo,, denotes a prime belonging to 

 the interval (|x, x) and satisfying the inequality 



Pn+1-Pn>pl, 



then the interval (pn, Pn+i) will contribute more than y^, and a 

 fortiori more than i (^xf, to a^'. Hence we obtain 

 1 (ia;)* (A {x) - h iix)) = (x^- i^+^), 



h(x)-h (i«) = (a;i-?*+^). I 



If we replace in the last relation x first by ^x, then by ^x, and so 

 on, and add together all the relations obtained in this way, we get 



Hence our theorem is proved. 

 Cambridge, 20 July, 1920. 



