We prove that the number of primes in an interval of length N is at most 2N Log N when N is large enough This is obtained through a sieving process which can be seen as a hybrid between the large sieve and the Selberg sieve and draws on what we call ”local models”

Abstract We prove that the number of primes in an interval of length N is at most 2N/(Log N +3.53) when N is large enough. This is obtained through a sieving process which can be seen as a hybrid between the large sieve and the Selberg sieve, and draws on what we call ”local models”. 1

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