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            <title xml:lang="en">Improving on the Brun-Titchmarsh Theorem</title>
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              <p>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".</p>
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