Dr Cramer, On the distribution of primes 



273 



It is interesting to remark that we may obtain by a very 

 trivial argument {viz. that the sum of all the h (x) A„'s which are 

 greater than p^ must be less than x + x") the evaluation 



h(x) = 0(x^-''), 

 but it seems impossible to improve this even by direct deduction 

 from the Prime Number Theorem. 



The proof of the theorem given here depends on the theory ot 

 the function le"', which 1 have studied in some recent papers*. 

 We denote here and in the sequel by p = i + 17 an arbitrary zero 

 of the function ^s), situated in the upper half of the plane ot 

 the complex variable s = o- + it. 



In order to prove the theorem, we shall require a set ot lemmas. 

 It seems convenient to remark that all the sets of points we shall 

 have to deal with in the proof consist of a finite number of finite 

 intervals (and perhaps a finite number of isolated points). Hence 

 their measure may be taken to be the measure in the elementary 

 sense. 



Lemma 1. If 



(f) (y, t)=t ey^K 



tve have 



r^^ \<f> (v,t)\dt= {v^ (\ogvf) 



Jx-2 



uniformly for x >2. 

 Proof We have 



and thus 



■a;+2 ^ ^ 1'^+^ 



y<v y'<v ■ X-2 



/ 2 \ 



< 1 1 Minf- rp4 



x-2 y^v y'^v- x-2 



The number of numbers y in the interval (y + v,ry + v + l) is 

 (log (7 + i/))t- It follows that our sum is 



o(t^os,)^o\ s (!^£(i±l)/.2£(|±i)+.,. 



\<V ' Ly<«-1 ^ 



= 0{v(^og,vy\ 



log (7 + [^-7]) 



[w-7] 



* l.c. (footnote i) ; Comptes rendus, t. 168, p. 539 ; and Mathematische Zeit- 

 schrift, Band 4, pp. 104 — 



I Landau, Handbuch, p. 337. 



