Improving on the Brun-Titchmarsh Theorem

We prove that the number of primes in an interval of length $N$ is at most $2N\mathrm{/}(\log N+3.53)$ 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”. This paper has been published in {\sc Acta Arithmetica, 2008, vol. 131, nb. 4, pages 351–366}.