Primes in tuples I | Annals of Mathematics

Abstract We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim infn→∞pn+1−pnlogpn=0. We will quantify this result further in a later paper.

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数学年刊(Annals of Mathematics)

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