Math 597e Prime
Number Theory, Spring 2008
Link
to Syllabus, Lecture Notes, Homeworks
and Solutions
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.
We 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 that
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.