272 Dr Cramer, On the distribution of primes 



On the distribution of primes. By H. Cramer, Stockholm. 



(Communicated by Prof. G. H. Hardy.) 



[Received 10 August 1920 : read 25 October.] 



Throughout the whole of this paper, I shall assume the truth of 

 the Riemann hypothesis concerning the roots of the Zeta function, 

 viz. ^(o- + it) 4" for <t>\. 



This being so, it is known that 



TT («) = Xi (a;) + (ic* log a?) (1 ) 



and Pn'^n log n (2), 



where ir {x) denotes the number of primes less than or equal to x, 

 andpn denotes the nth prime. The last relation is independent of the 

 Riemann hypothesis. But very little is known as to the behaviour 

 of the difference 



^n=Pn + i-Pn 



between two successive primes, for large values of n. It follows 

 from the " Prime Number Theorem " (1) or (2) that 



log w ~ logp„ 



and from (1) that 



An=0{p«*log2^„). 



I have recently shown* that the last relation may be replaced by 



K = 0{p^\ogpn). 



So far as I know, this is all that is actually known about A^- 

 It IS very probable that A^= 2 for an infinity of values of n; but 

 this has not yet been proved, and it has not even been proved 

 that A,,<ilogp^ or A^ > 2 log^„ for an infinity of values of n 

 Ihe object of the present paper is to prove the following theorem 

 which gives an upper limit for the frequency of certain large values 

 of A„. *= 



Theorem. Let h {x) be the number of primes »„ < x satis fvina 

 the inequality ~ ^ ^ if 



Vn + i-pn>pl, 

 where 0<k ^^. Then 



h{x)=o(x'-^'+') i 



for every positive e. j 



