**Recent work of C.
Yildirim and D. Goldston**

** What are the
shortest intervals between consecutive prime numbers? The twin prime conjecture, which
asserts that infinitely often is one of the oldest problems; it is difficult
to trace its origins. **

** In the
1960's and 1970's sieve methods developed to the point where the great Chinese
mathematician Chen was able to prove that for infinitely many primes the number is either
prime or a product of two primes. However the well-known ``parity problem'' in sieve
theory prevents further progress. **

** What can
actually be proven about small gaps between consecutive primes? A restatement of the prime
number theorem is that the average size of is where denotes the th prime. A consequence is
that **

** In 1926,
Hardy and Littlewood, using their ``circle method'' proved that the Generalized Riemann
Hypothesis (that neither the Riemann zeta-function nor any Dirichlet L-function has a zero
with real part larger than 1/2) implies that . Rankin improved this (still assuming
GRH) to . In 1940 Erdös, using sieve methods, gave the first unconditional proof that . In
1966 Bombieri and Davenport, using the newly developed theory of the large sieve (in the
form of the Bombieri - Vinogradov theorem) in conjunction with the Hardy - Littlewood
approach, proved unconditionally, and then using the Erdös method they obtained
. In 1977, Huxley combined the Erdös method and the
Hardy - Littlewood, Bombieri - Davenport method to obtain .
Then, in 1986, Maier used his discovery that certain intervals contain a factor of more
primes than average intervals. Working in these intervals and combining all of the above
methods, he proved that , which was the best result until now. **

** Dan Goldston
and Cem Yildirim have a manuscript which advances the theory of small gaps between primes
by a quantum leap. First of all, they show that . Moreover, they can prove that for
infinitely many the inequality **

**holds. **

** Goldston's
and Yildirim's approach begins with the methods of Hardy-Littlewood and Bombieri -
Davenport. They have discovered an extraordinary way to approximate, on average, sums over
prime -tuples.
We believe, after work of Gallagher using the Hardy-Littlewood conjectures for the
distribution of prime -tuples, that the prime numbers in a short interval are distributed like a Poisson random variable with parameter . Goldston and
Yildirim exploit this model in choosing approximations. They ultimately use the theory of
orthogonal polynomials to express the optimal approximation in terms of the classical
Laguerre polynomials. Hardy and Littlewood could have proven this theorem under the
assumption of the Generalized Riemann Hypothesis; the Bombieri - Vinogradov theorem allows
for the unconditional treatment. **

** This new
approach opens the door for much further work. It is clear from the manuscript that the
savings of an exponent of 1/9 in the power of is not the best that the method will
allow. There are (at least) two possible refinements. One is in the examination of lower
order terms that arise in his method. Can they be used to enhance the argument? The other
is in the error term Gallagher found in summing the ``singular series'' arising from the
Hardy-Littlewood -tuple conjecture. There is reason to believe that this error term can be
improved, possibly using ideas in recent work of Montgomery and Soundararajan (``Beyond
Pair - Correlation''.) **

** It is not
clear just how far this method can be pushed and what other problems might be attacked
using his new ideas; at this point we can't rule out developments that would even approach
the centuries old twin prime problem. What is clear is that a monumental barrier which has
impeded progress for at least the last 80 years has been broken down. **

** Note:
Actually this work is astonishing in another regard. They have actually proven that for
any fixed number the inequality **

**holds for infinitely many . Those
familiar with work on large gaps between primes will recall that in 1977 Helmut Maier
burst onto the analytic number scene with his tour de force proof that the largest gaps
known to hold for two consecutive primes could be proven for each gap of consecutive gaps for any
fixed .
Goldston and Yildirim have achieved a similar sort of result for consecutive small gaps at
the same time that they have demolished all previous records for one small gap. **