d******s
发帖数: 180 | 1 http://aimath.org/news/primegaps70m/
第一段就很有料，四月后期投稿，五月中旬便接受，对于一篇55页的paper来说速度惊
人。
Zhang's Theorem on Bounded Gaps Between Primes
by Dan Goldston
In late April 2013 Yitang Zhang of the University of New Hampshire submitted
a paper to the Annals of Mathematics proving that there are infinitely many
pairs of primes that differ by less than 70 million. The proof of this
amazing result was verified with high confidence by several experts in the
field and accepted for publication. A public slightly revised version of the
paper should be available shortly.
Before describing the proof, we make the caveat: As with any major result,
the proof needs to be widely examined for errors before it can be completely
accepted as correct. In this case the 55 page paper is detailed and precise
, and the probationary period will probably be a few months.
Zhang's theorem is a huge step forward in the direction of the twin prime
conjecture. We now know for the first time that there are actually
infinitely many pairs of primes that differ by some fixed number. The proof
does not specify any specific number, only that there is at least one that
is less than 70 million. (We actually expect that every even number will
occur as the difference of two primes infinitely often.) Since the average
spacing of primes around x is logx which grows to infinity slowly, this
sequence of pairs defy this isolating trend. If not twins, then certainly
siblings.
Zhang's proof makes use of a modified version of the 2005 method of Goldston
, Pintz, and Yildirim (GPY). Consider the tuple (n+h1,n+h2,…,n+hk), where
the hi's are specified integer shifts, and n runs through the positive
integers. The Hardy-Littlewood k-tuple conjecture states that there are
infinitely many n where every component of the tuple is simultaneously prime
, except for hi's where this is clearly impossible. Thus (n,n+1) is only a
prime tuple for (2,3) since one of n or n+1 is even and divisible by 2.
Similarly (n,n+2,n+4) only is a prime tuple for (3,5,7) since one component
is divisible by 3. On the other hand (n,n+2) and (n,n+2,n+6) have no such
obstruction and we expect them to be prime tuples infinitely often; the
former giving the twin primes. A tuple without the obstruction that some
prime always divides one of its components is called admissible. How GPY
works is to apply a sieve, actually the Selberg sieve, to a given tuple when
n runs over positive integers ≤x . Roughly speaking, the sieve removes the
tuples where any component has a small prime factor and leaves only tuples
where all the components have only large prime factors. Therefore the tuples
left have relatively few prime factors distributed among their components,
and we obtain a sequence where we are likely to find primes close together.
If for example we had a 10-tuple where the components together have at most
18 prime factors this would force at least two components of the tuple to be
prime and we would have produced two primes whose difference is at most the
width of the tuple. Sieves never work this well or we would have solved
most of the problems concerning primes long ago. Despite this, the Selberg
sieve applied to larger k-tuples is still highly effective at producing "
almost prime" tuples. We need to use some further idea in order to detect
the primes in the tuple, and what we use is the level of distribution of
primes in arithmetic progressions.
The prime number theorem says that the number of primes ≤x is
asymptotically x/logx as x→∞. This was proved in 1896 by Hadamard and de
la Vallèe-Poussin, and in 1899 de la Vallèe-Poussin extended the result to
prove the prime number theorem for arithmetic progressions. Thus for
example if we look at the three arithmetic progressions modulo 3 we have no
primes except 3 in 3,6,9,12,… but asymptotically half of all the primes
will occur in 1,4,7,10,… and the other half occur in 2,5,8,11,…. The
principle is that for the q progressions modulo q the primes will flow
evenly into the progressions that allow primes, namely those for which there
is no prime dividing every term in the progression. There are ϕ(q)
such progressions, where ϕ(q) is the Euler phi-function equal to the
number of integers a, 1≤a≤q for which a and q are relatively primes. Hence
each progression gets asymptotically 1/ϕ(q)×x/logx primes. For
applications we usually need to have q be a function of x, and for this our
knowledge is extremely limited. For example, if q=xα then there is no α>0
known for which the asymptotic formulas are always true. Assuming the
Generalized Riemann Hypothesis (GRH) this is know for any α<1/2. A major
advance in the field occurred in 1965 when Bombieri and independently
Vinogradov proved that the asymptotic formula is true for α<1/2 for "almost
all" progressions. Since in many applications we break our expression up
over lots of progressions, the Bombieri-Vinogradov Theorem has exactly the
same strength as GRH for these applications. If in place of 1/2 we assume
that almost all progressions satisfy the prime number theorem for arithemtic
progressions for q=xα and α<θ then we say the primes have level of
distribution θ. It is conjectured that the primes have level of
distribution 1, which is called the Elliott-Halberstam conjecture, but it
appears very difficult to go beyond the know level of distribution 1/2, and
GRH does not help with this either.
With this preparation we return to GPY and take the sequence of almost prime
tuples produced by the Selberg sieve and freeze (or twist) one of the
components to be a prime. We can compute this proportion because the sieving
is implemented through multiples of any divisors of the other components,
and we then count how often the frozen prime component occurs in these
arithemtic progressions. We can freeze individually each of the k components
to be a prime and we get the same result in each case. If the amount we get
when we freeze one component to be a prime is greater than 1/k times the
contribution of all the almost prime tuples, then the k events where one
component is frozen to be a prime cannot all be disjoint from each other and
there must be some overlap where two of the components are primes
simultaneously. This produces two primes in the given tuple. Therefore the
success of the method relies on the ratio of the almost primes tuples
produced by the Selberg sieve to the almost prime tuples produced where one
component is a prime. The Selberg sieve coefficients need to be chosen
appropriately, and while the optimal choice is not clear, it isn't too hard
to make a natural and very good choice. In evaluating the twisted sum the
size of the numbers we use in sieving is limited by the level of
distribution, and therefore the sieving is improved by a larger level of
distribution.
The result of this procedure obtained in GPY is the following. If the level
of distribution θ>1/2 then you actually win and obtain two primes in your
tuple for sufficiently large k. Thus you get bounded gaps between primes
under this unproved assumption. If we assume a level of distribution 1, the
Elliot-Halberstam Conjecture, then we get two primes infinitely often in
every admissible 6-tuple. The smallest admissible 6-tuple is (n,n+4,n+6,n+10
,n+12,n+16) which gives pairs of primes with difference ≤16. Since we only
know level of distribution 1/2, GPY fails to get 2 primes in any tuple
unconditionally. However not all is lost, and by separate arguments one can
still produce two primes significantly closer than the average spacing
between primes. However these small gaps between primes are unbounded and
grow to infinity.
The most tantalizing question left by GPY was whether one could edge over
the level of distribution barrier at θ=1/2 to get bounded gaps between
primes. Soundararajan showed that improving the choices in the Selberg sieve
could not succeed, and that shifted the question onto the level of
distribution error terms that arise in the method. In the early 1980's
Fouvry and Iwaneiec were able to obtain results connected to primes in
arithmetic progressions with a level of distribution greater than 1/2.
During 1985-90 Bombieri, Friedlander, and Iwaniec wrote three well-known
papers where they obtained many results which go beyond level of
distribution 1/2. In particular these papers obtain results beyond what GRH
implies, and require estimates from the spectral theory of automorphic forms
. Their results in some cases corresponding to a level of distribution 4/7.
One limitation of these methods is that they do not apply directly to sums
of absolute value of the errors terms in the progressions, but rather to
weighted sums of the error terms, where the weights have a fair degree of
flexibility. Since in applications the error terms also come weighted by
various arithmetic functions, these new results have found a number of
important uses. For example, Brian Conrey's proof that more than 2/5's of
the zeros of the Riemann zeta-function are on the half-line uses this
analysis. In the case of GPY however, despite attempts by a number of
mathematicians, no one was able use these techniques to deal with the error
terms generated by the method.
Now Zhang has found how to break this barrier. The first step is a
modification in the GPY method. Zhang shows that in using the Selberg sieve
one can restrict the sieving to divisors with no large prime factors. This
of course decreases the effectiveness of the sieve, but in GPY he shows that
this decrease has a surprisingly small effect on the method. Next, he
increase the sieving range corresponding to a level of distribution 1/2+1/
584. This turns out to be large enough to overcome the loss from removing
the large prime factors above and give bound gaps between primes, provide
all the error terms can be controlled. This leads to a very intricate
analysis using the methods of Bombieri, Friedlander, and Iwaniec. Of crucial
importance is that the divisors in the sieving have no large prime factors
and therefore may themselves be factored into factors of various sizes with
considerable flexibility. Ultimately it is shown that all the error terms
are controlled and the result follows. It should be noted that this does not
provide a proof that the primes themselves have a level of distribution
greater than 1/2 according to our definition of level of distribution above.
It is too early to see how Zhang's method may be used in other applications.
Zhang's proof ends up proving that every admissible k-tuple with k=3,500,000
contains 2 primes. It is not easy to find the smallest such admissible
tuple, but a good choice is to take the shifts hi to be the first k primes
that are larger than k. Using a math package and rounding up to the nearest
10 million gives the size 70,000,000 for the width of the tuple and thus the
difference between the two primes. (Actually 60,000,000 seems to work here.
) This number has attracted attention, but it has no intrinsic significance.
In working out a complicated proof, one often uses whatever numbers make
the argument work out reasonably simply. No doubt much smaller numbers will
be found in later papers of greater complexity as the limits of the method
are explored. At the moment it is hard to predict even the rough size of the
ultimate answer by this method. On the other hand, it would be nice to have
simpler and shorter proofs which make no attempt to get any specific number. |
