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.
Prime numbers are of interest to anyone with curiosity who has learned to multiply. One sees that the counting numbers either occur as products of smaller counting numbers, for example 9=3x3, or else occur for the first time by themselves: 2, 3, 5, 7, 11, . . . . These are the prime numbers. Despite their simple origin, many basic questions about prime numbers have never been answered. For example, 5 and 7 are two consecutive primes that differ by 2, as are 11 and 13. These are called twin primes. By computation we can easily find large twin primes, and we suspect that this will continue forever. This is the twin prime conjecture: There are infinitely many twin primes. This conjecture has never been proved, and many mathematicians see little hope of proving it for a long time. In 2005 Pintz, Yildirim and I found a new method that is a small step in the direction of proving the twin prime conjecture. We proved that there are infinitely many primes that are very close to their next prime neighbor, where close means closer than any fraction of the average distance to a neighbor. Further, our method actually gives that infinitely many primes are at a distance of 16 or less from their nearest neighbor provided that a separate conjecture that primes are very evenly distributed in arithmetic progressions holds. The first outcome of the current project is that Pintz, Yildirim and I published most of our work from 2005 and it was widely distributed. A second outcome is a refinement of our result from 2005: we proved that the number of primes with a nearby neighbor forms a positive proportion of all the primes. Specifically, given any y, we can find an x such that at least x% of all primes have a neighbor within y% of the average distance between primes. For example, we expect that just about 1% of the primes have a neighboring prime closer than 1% of the average distance between primes. We can not prove this, but we can prove that at least 10^{-10^18} % of the primes have this property. This may seem like a ridiculously poor result, but it is the first foot in a door that has never been opened before. Another part of the current project is concerned with jumping champions, which are the most commonly occuring distances between consecutive primes for primes up to x. The know jumping champions are 1, 2, 4, and 6. As soon as x is larger than 1000, 6 is the jumping champion, and this continues as far as the latest computation where x is 10^15. However, it is conjectured that 6 will eventually lose to the next jumping champion 30, which will eventually lose to the next jumping champion 210, and so on through the primorials 1, 2, 6, 30, 210, 2310, . . . = 1, 2, 2x3, 2x3x5, 2x3x5x7, 2x3x5x7, 2x3x5x7x11 . . . . In joint work with Andrew Ledoan, we proved that a quantitative form of the prime twin conjecture together with a related conjecture for prime triples implies that large enough jumping champions will jump through the primorials. These conjectures are very difficult, but our result does move the Jumping Champion problem into the realm of standard conjectures for primes. We failed to prove anything unconditional about jumping champions, such as 6 will not always be the jumping champion, or even that 2 will not be the final jumping champion. The work done in this project was curiosity driven without regard to future applications. However, over and over it has been found that work which develops the structure and improves the tools available in a specific area of knowledge pays off in totally unexpected and valuable applications. Time will tell whether that is the case here.
