The object of this course is to describe and explore the ideas underlying the very recent major developments in the theory of prime numbers by Goldston, Pintz and Yıldırım, and by Green and Tao. This is particularly timely as Professor Tao will be the 2008 Marker Distinguished Lecturer during November 2008. A prerequisite is some basic knowledge of the distribution of primes into arithmetic progressions such as is often covered in Math 571 or Math 572, or occasionally in Math 567 or Math 568. Alternatively, some acquaintance with a standard text on the subject, such as in Davenport’s Multiplicative Number Theory, or Montgomery and Vaughan’s Multiplicative Number Theory I. Classical Theory, §11.3, would suffice.
know from the prime number theorem that if is the -th
prime in order of magnitude, then has average value . On the other hand, if the twin prime
conjecture is true, then holds for infinitely many . In view of our inability to prove the twin
prime conjecture it is natural to study
Until recently the best estimate for λ, following work of a
host of famous mathematicians, including Hardy and Littlewood,
Erdős, Rankin, Ricci, Davenport and Bombieri ( ), and Huxley, is Maier’s . In a remarkable piece of work, using only
classical ideas, Goldston,
Pintz and Yıldırım have established
Moreover on the assumption of a conjecture concerning the
distribution of primes into arithmetical progressions, which is widely believed,
they are able to show that there is an absolute constant such that for infinitely many
It is not so difficult to find arithmetic progressions in the primes. Here are some examples.
It was conjectured for at least a century that there are arbitrarily long arithmetic progressions of primes. In 2004 this was established in a major piece of work by Green and Tao. The proof brings together ideas from several areas. In one part of the argument, use is made of a theorem of Goldston and Yıldırım which also plays a role in the work of Goldston, Pintz and Yıldırım described above.
In slightly more precise language the problem is to find, for arbitrarily large , primes and a positive integer which satisfy the simultaneous equations . Thus there are unknowns and equations. Similar situations with have long had a solution. For example, following seminal work of Hardy and Littlewood, Vinogrodaov showed in 1937 that for all large odd there are primes such that (one equation and three unknowns). A variant of the Vinogradov method can be used to show, for example, that for any fixed even integer there are infinitely many primes such that .
The topics covered in this course will include
· The large sieve.
· Bombieri’s theorem on primes in arithmetic progression, which tells us that the generalized Riemann hypothesis is true on average.
· The Selberg sieve.
· The Vinogradov three primes theorem and a proof that almost all even natural numbers are the sum of two primes.
· The Goldston, Pintz and Yıldırım proof that
· Some aspects of the Green, Tao work.