Dr Cramer, On the distribution of primes 275 



Proof. In a recently published paper*, I have proved the 



formula 



r..\ ^ >-i){^+'2/) /I 



^ ! « ((t - Tog ,;y ^f )'!-"^^ ^^1^ ^ " Vi 



where 3/ may denote any positive decreasing function of t which 

 tends to zero as t tends to infinity. On the Riemann hypothesis, 

 and assuming y to be the function of Lemma 2, this formula 

 reduces to 



^Tw((^-logw)^ + y') 

 = ^ _ ^ e- i^ (^1 + 0)) S cos (7« - li/ - arg (^ f iy)) e-yv 



= ^_^e-*^(l + 0(f))(Scos7«e-v^ + 0(2/2e-v^)) + 0^^ 



But we have 



^e-yv = (i log ne-'"J\ = I- log - j . 



Hence 



^i ^>) ='^(l-2e-'i'tcosyte-yA + o(] 



^in{{t-\ogny + y') t\ ^ ' J \t 



By the definition of the set S^, we have 



\2e~^*tcosyte~'^^\< i 



for all values of t belonging to the complementary set So,. Since S^ 

 is contained in the interval (a;-2, x + 2), we conclude that 

 ?> ''? (^) _ "^ + ^ 



7 n{{t^]ognf+f) ~ Iy' ' 

 where ] ^ ] < 1 for all sufficiently large values of x. 

 Lemma 4. We denote by f {v) the function 

 f{v)=^ v{n) 



introduced by Riemann, and we consider the interval {x-2,x + 2) 

 of the tiuo preceding lemmas. The set of points t belonging to this 

 interval such that 



2 v{n)=f{e'^y)-f{e')>{e' 



is, for all sufficiently large values of x, a subset of the set S^ of 

 Lemma 2, so that its measure is of the form 



oix'e'"^'V. 



Z.c, footnote*, p. 272. 



18—5 



