Primes in Tuples I

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, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{p_{n+1}-p_n}{\log p_n}=0.$$ This last result will be considerably improved in a later paper.