274 Dr Cramer, On the distribution of primes 



iformly in x. Hence the 

 Schwarz's inequality. 



Lemma 2. Let us put 



uniformly in x. Hence the truth of the lemma follows by the use 

 of Schwarz's inequality. 



y^-t' 



-at 



and consider the interval x-2^t^x + 2, where a; > 2 and ^ < o < !• 

 Then the set S^; of points t belonging to this interval such that 



I 2 cos ryt e-"" ' ^ le*' 



I y 



is of measure M„ such that 



Proof We have 



^co8r^te-yy\<\^(e''".e-A = y T <},(v,t) e''^'^ dv 



y 1 1 V ^ / Jo 



= 2/ \(f)(v,t)\ e-^ydv. 

 Jo 



It follows by Lemma 1 that, if we denote by y^ and y, the 

 values of y at the points x-2 and a; + 2 respectively, 



Scos7^ e-yy \dt=Oi\ y^v^ {log vfe-'>'y^dv) 



x-2 



= 0(y,2/r^(logl) 



= {x'e-'^). 



Thus the measure of the set of points t belonging to the interval 

 of integration, such that 



X cos <yt e "^^ 



^le' 



J4 = 0(a;V*"^e~*^) 

 = We 2 



must be of the form 



Lemma 3. In the set S^, complementary to the set S^ of 

 Lemma 2, we have 



X ^(>^) 'rr + O 



in((t-\ogny + y^) ty ' 

 where \0\<1 for all sufficiently large values of x. Here 97 (n) 

 denotes the arithmetical function defined for integral values of 

 n^l by ^ c J 



T){n) = ~ [71 = p'^^ p prime), 

 v{n) = (otherwise). 



