Gaps Between Primes

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In 2005 the investigator, jointly with Janos Pintz and Cem Yildirim introduced a method that proved for the first time that there exist infinitely often pairs of prime numbers whose difference is smaller than any fraction of the average spacing between primes. The method is especially interesting because it demonstrates that the distribution of primes in arithmetic progressions contains more information on the behavior of primes than had previously been recognized. In particular, if the primes are well distributed in "thin" arithmetic progressions then one can prove that there are always pairs of primes with difference 16 or less. This conditional result shows that the twin prime conjecture - that there are infinitely many primes differing by 2, is approachable with this type of information. More recently the PI in joint work with Yildirim and Pintz and Sid Graham have been working on applying our method to numbers with a specific number of prime factors, especially numbers which have exactly two prime factors. The main goal of this project is to further develop our method to extract the best possible quantitative information on small gaps between primes, and in addition to investigate applications to other problems in number theory. The investigator is also interested in questions related to the use of explicit formulas to study zeros of the Riemann zeta-function.This project is concerned with prime numbers, an ancient subject extending back to the Greeks and up to the present with many important applications in computer science and cryptography. Despite the simplicity of how they arise, prime numbers offer some of the most challenging and difficult problems in mathematics. Many if not most mathematicians judge the most famous and important unsolved problem in mathematics to be the Riemann Hypothesis, and this is equivalent to the primes being distributed in a fairly regular distribution. More generally, the field of number theory has applications throughout mathematics and fields that make use of mathematics.