i****g
发帖数: 3896 | 1 http://blog.sina.com.cn/s/blog_c24597bf0101b871.html
致谢：I would like to thank Prof. Shing-Tung Yau for suggesting the title of
this article, Prof. William Dunham for information on the history of the
Twin Prime Conjecture, Prof. Liming Ge for biographic information about
Yitang Zhang, Prof. Shiu-Yuen Cheng for pointing out the paper of
Soundararajan cited in this article, Prof. Lo Yang for information about
Chengbiao Pan quoted below, and Prof. Yuan Wang for detailed information on
results related to the twin prime conjecture, and Prof. John Coates for
reading this article carefully and for several valuable suggestions.
What is mathematics? Kronecker said, “God made the integers, all the rest
is the work of man.”What makes integers? Prime numbers! Indeed, every
integer can be written, essentially uniquely, as a product of primes. Since
ancient Egypt (around 3000BC), people have been fascinated with primes.
More than two thousand years ago, Euclid proved that there are infinitely
many primes, but people observed that primes occur less and less frequently.
The celebrated Twin Prime Conjecture says the external exceptions can occur,
i.e., there are infinitely many pairs of primes with gaps equal to 2. The
first real breakthrough to this century old problem was made by the Chinese
mathematician Yitang Zhang: There are infinitely many pairs of primes each
of which is separated by at most 70 millions.
Prime numbers
The story of primes is long and complicated, and the story of Zhang is
touching and inspiring. In some sense, the history of primes gives an
accurate snapshot of the history of mathematics, and many major
mathematicians have been attracted to them.
After Euclid, the written history of prime numbers lay dormant until the
17th century
when Fermat stated that all numbers of the form $2^{2^n} + 1$ (for natural
numbers $n$) are prime. He did not prove it but he checked it for $n =1,
cdots, 4$. Fermat's work motivated Euler and many others. For example,
Euler showed that the next Fermat number $2^{2^5}+1$ is not prime. This
shows the danger of asserting a general statement after few experiments.
Euler worked on many different aspects of prime numbers. For example, his
correspondences with Goldbach in 1742 probably established the Goldbach
conjecture as a major problem in number theory. In his reply, Euler wrote
“every even integer is a sum of two primes. I regard this as a completely
certain theorem, although I cannot prove it.”
Distribution of primes is a natural and important problem too and has been
considered by many people. Near the end of the 18th century, after extensive
computations, Legendre and Gauss independently conjectured the Prime Number
Theorem： As $x$ goes to infinity, the number $pi(x)$ of primes up to $x$
is asymptotic to $x/ln(x)$. Gauss never published his conjecture, though
Legendre did, and a younger colleague of Gauss, Dirichlet, came up with
another formulation of the theorem. In 1850, Chebyshev proved a weak
version of the prime number thorem which says that the growth order of the
counting function $pi(x)$ is as predicted, and derived as a corollary that
there exists {em a prime number between $n$ and $2n$ for any integer $n
geq 2$.} (Note that the length of the period goes to infinity).
Though the zeta function was used earlier in connection with primes in
papers of Euler and Chebyshev, it was Riemann who introduced the zeta
function as a complex function and established close connections between the
distribution of prime numbers and locations of zeros of the Riemann zeta
function. This blending of number theory and complex analysis has changed
number theory forever, and the Riemann hypothesis on the zeros of the
Riemann zeta function is still open and probably the most famous problem in
mathematics.
In his 1859 paper, Riemann sketched a program to prove the prime number
theorem via the Riemann zeta function, and this outline was completed
independently by Hadamard and de la Vall'ee Poussin in 1896.
One unusual (or rather intriging, interesting) thing about primes is that
they exhibit both regular and irregular (or random ) behaviors. For
example, the prime number theorem shows that its overall growth follows a
simple function, but gaps between them are complicated and behave randomly
(or chaotically). One immediate corollary of the prime number theorem is
that gaps between primes go to infinity on average (or the density of primes
among integers is equal to zero). Understanding behaviors of these gaps is a
natural and interesting problem. (Note that in some ways, this also reflects
the difficulty in handling the error terms in the asymptoptics of the
counting function $pi(x)$ of primes, which are related to the zeros of the
Riemann zeta function as pointed out by Riemann).
Gaps in primes
What is the history of study on gaps of primes? In 1849, Polignac
conjectured that every even integer can occur as gaps of infinitely many
pairs of consecutive primes, and the case of 2, i.e., the existence of
infinitely many pairs of twin prime pairs, is a special case. The twin prime
number in the current form was stated by Glaisher in 1878, who was Second
Wrangler in 1871 in Cambridge, a number theorist, the tutor of a famous
philosopher Ludwig Wittgenstein, and President of the Royal Astronomical
Society. After counting pairs of twin primes among the first few millions of
natural numbers, Glaishe concluded: “There can be little or no doubt that
the number of prime-pairs is unlimited; but it would be interesting, though
probably not easy, to prove this.”
These were the first known instances where the twin prime conjecture was
stated. Given the simple form and naturalness of the twin prime conjecture,
it might be tempting to guess that this question might be considered by
people earlier. It might not be a complete surprise if it were considered by
the Greeks already. But according to experts on the history of mathematics,
in particular on the work of Euler, there was no discussion of twin primes
in the work of Euler. Since Euler was broad and well versed in all aspects
of number theory, one might conjecture that Polignac was the first person
who raised the question on twin primes.
This easily stated conjecture on twin primes has been attacked by many
people. Though the desired gap 2 is the dream, any estimates on them smaller
than the obvious one from the prime number theorem is valuable and
interesting, and any description or structure of distribution of these gaps
is important and interesting as well1.
Many partial and conditional results have been obtained on sizes of gaps
between primes, and there are also a lot of numerical work listing twin
prime pairs. Contributors to this class of problems include Hardy,
Littlewood, Siegel, Selberg, Rankin, Vinogradov, Hua, Erd"os, Bombieri,
Brun, Davenport, Rademacher, R'enyi, Yuan Wang, Jingrun Chen, Chengdong
Pan, Friedlander, Iwaniec, Heath-Brown, Huxley, Maier, Granville,
Soundararajan et al. Indeed, it might be hard to name many great analytic
number theorists in the last 100 years who have not tried to work on the
twin prime conjecture directly or indirectly. Of course, there are many
attempts by amateur mathematicians as well.
One significant and encouraging result was proved in 2009 by Goldston,
Pintz, and Yildirim. In some sense, they started the thaw of the deep
freeze. They showed that though gaps between primes can go to infinity, they
can be exceptionally small. This is the result of “culminating 80 years of
work on this problem” [2, p.1].
Under a certain condition called Elliott-Halberstam conjecture on
distribution of primes in arithmetic progressions, they can prove that there
are infinitely many pairs of primes with gaps less than 16. Though this
condition might be “within a hair's breadth” of what is known [1, p. 822],
it seems to be hard to check.
In [1, p. 822], they raised “Question 1. Can it be proved unconditionally
by the current method that there are, infinitely often, bounded gaps between
primes?”
How to make use of or improve such results? Probably this was the starting
point for Zhang. But problems of this kind are hard [1, p. 819]: “Not
only is this problem believed to be difficult, but it has also earned the
reputation among most mathematicians in the field as hopeless in the sense
that there is no known unconditional approach for tackling the problem.”
Indeed, difficulties involved seemed insurmountable to experts in the field
before the breakthrough by Zhang. According to Soundararajan [4, p. 17]:
“First and most importantly, is it possible to prove unconditionally the
existence of bounded gaps between primes? As it stands, the answer appears
to be no, but perhaps suitable variants of the method will succeed.”
The most basic, or the only method, to study prime numbers is the sieve
method. But there are many different sieves with subtle differences between
them, and it is an art to design the right sieve to attack each problem.
Real original ideas were needed to overcome the seeming impasse. After
working on the problem for three years, one key revelation occurred to Zhang
when he was visiting a friend's house in July 2012, and he solved Question 1
in [1]. In some sense, his persistence and confidence allowed him to succeed
at where all world experts failed and gave up.
On May 13, 2013, Zhang gave a seminar talk at Harvard University upon the
invitation of Prof. Shing-Tung Yau. At the seminar, he announced to the
world his spectacular result [5]: There exist infinitely many pairs of
primes with gaps less than 70 millions.
This marks the end of a long triumphant period in analytic number theory and
could be the beginning of a new period, leading to the final solution to the
twin prime conjecture.
Zhang's career
Zhang's academic career is a mingling of the standard and the nonstandard,
maybe similar to prime numbers he loves. He went to Beijing University in
1978 and graduated from college as the top student in 1982.2 Then in 1982—
1985, he continued to study for the Master degree at Beijing University
under Chengbiao Pan and hence was an academic grandson of L.K. Hua3. After
receiving a Ph. D. degree at Purdue University in 19924, he could not get a
regular academic job and worked in many areas at many places including
accounting firms and fast food restaurants. But mathematics has always been
his love. From 1999 to 2005, he acted as a substitute or taught few courses
at University of New Hamsphire. From 2005 to present, he has been a
lecturer there and is an excellent teacher, highly rated by students. In
some sense, he has never held as a regular research position in mathematics
up to now. It is impressive and touching that he has been continuing to do
research on the most challenging problems in mathematics (such as the zeros
of the Riemann zeta function and the twin prime conjectures) in spite of the
difficult situation over a long period. His persistence has paid off as in
the Chinese saying: 皇天不负有心人 (Heaven never disappoints those
determined people, or Heaven helps those who help themselves!)
His thesis dealt with the famous Jacobian conjecture on polynomial maps,
which is also famous for many false proofs and is still open. After
obtaining his Ph. D. degree and before this breakthrough on twin primes,
Zhang published only one paper, ``On the zeros of $zeta'(s)$ near the
critical line" [3] in the prestigous Duke Journal of Mathematics, which
studies zeroes of the Riemann zeta function and its derivative and gaps
between the zeros. In 1985, Zhang published another paper on zeros of the
Riemann zeta function in Acta Mathematica Sinica, one of the leading
mathematics journals in China.
It is perhaps helpful to point out that spacing of these zeros and twin
primes are closely related [4, p. 2]: ``Precise knowledge of the frequency
with which prime pairs $p$ and $p + 2k$ occur (for an even number $2k$) has
subtle implications for the distribution of spacings between ordinates of
zeros of the Riemann zeta-function.... Going in the other direction, weird
(and unlikely) patterns in zeros of zeta-like functions would imply the
existence of infinitely many twin primes."
In the current culture of mass production of everything, probably one
sobering question is how much one should or can really produce. (This
reminds one the famous short story by Tolsty, ``How much land does a man
need?" One can also replace “land” by other attractive items, and “much”
by “many”.)
Is counting papers and number of pages an effective criterion? Probably one
should also keep in mind that the best judgment on everything under the Sun
is still the time!
Maybe not everyone is familiar with the Riemann hypothesis, but every
student who has studied calculus will surely have heard of Riemann and his
integrals. Many mathematicians will agree that Riemann is one of the
greatest mathematicians, if not the greatest mathematician, in the history.
But it is probably less known that in his life time, Riemann only published
5 papers in mathematics together with 4 more papers in physics. (Galois
published fewer papers in his life time, but he never worked as a full time
mathematician and died at a very young, student age.)
Primes are not loners
The concept of primes is also a sentimental one. Primes are lonely numbers
among integers, but for some primes (maybe also for some human beings), one
close partner as in twin prime pairs is probably enough and the best. This
sentiment is well described in the popular novel, “The Solitude of Prime
Numbers '' by Paolo Giordano. Let us quote one paragraph from this book:
Primes “are suspicious, solitary numbers, which is why Mattia [the hero of
the novel] thought they were wonderful. Sometimes he thought that they had
ended up in that sequence by mistake, that they'd been trapped, like pearls
strung on a necklace. Other times he thought that they too would have
preferred to be like all the others, just ordinary numbers, but for some
reasons they couldn't do it....among prime numbers, there are some that are
even more special. Mathematicians call them twin primes: pairs of prime
numbers that are close to each other, almost neighbors, but between them
there is always an even number that prevents them from touching.”
Zhang reminds one of the heroes of several generations of Chinese students,
Jingrun Chen, and his work on the famous Goldbach conjecture. Chen and Zhang
both worked persistently and lonely on deep problems in number theory, and
they both brought glory to China, in particular, to the Chinese
mathematics community.
Of course, the story of Jingrun Chen is well-known to almost every Chinese
student (young now and then). For a romantic rendition of a mathematician's
pursuit of the Goldbach conjecture, one might enjoy reading the book Uncle
Petros and Goldbach's Conjecture: A Novel of Mathematical Obsession by
Apostolos Doxiadis. (Incidentally, the hero in this novel, Uncle Petro,
published only one paper after his Ph. D. degree and before he switched to
work on the Goldbach conjecture.)
References
[1] D. Goldston, J. Pintz, C. Yldrm, Primes in tuples. I., Ann. of Math.
(2) 170 (2009), no. 2, 819--862.
[2] D. Goldston, J. Pintz, C. Yldrm, Primes in tuples. II., Acta Math.
204 (2010), no. 1, 1--47.
[3] Y. Zhang, On the zeros of $zeta'(s)$ near the critical line, Duke
Math. J. 110 (2001), no. 3, 555--572.
[4] K. Soundararajan, Small gaps between prime numbers: the work of
Goldston-Pintz- Yldrm, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1-
-18.
[5] Y. Zhang, Bounded gaps between primes, preprint, 2013, 56 pages.
1 The problem on spacing between prime numbers is one of several problems
concerning spacing in naturally occurring sequences such as zeros of the
Riemann zeta-function, energy levels of large nuclei, the fractional parts
of $sqrt{n}$ for $ n in mathbf N$. One question asks whether the spacings
can be modelled by the gaps between random numbers (or eigenvalues of
randomly chosen matrices), or whether they follow other more esoteric laws.
2 This seems to the unanimous opinion of former students from Beijing
University who knew him.
3 Though Chengbiao Pan was not a student of Hua formally, but the influence
of Hua on Pan was huge and clearly visible. According to Prof. L. Yang,
"Chengbiao Pan was a undergraduate student in PKU (1955-1960). He worked,
after 1960, in Beijing Agriculture Machine College, and became a professor
of Beijing Agricultute University in the eighties, after his college was
combined in this university. Though Pan was the professor of Beijing
Agriculture University, he spent most time in PKU. Pan was a student of
Prof. Min, but not studying the number theory. Instead of, Pan studied the
generalized analytic functions (the Russian mathematician Vekywa and L.
Bers), since Prof. Min had to change his field to this in 1958.) I believe
that Pan's knowledge and ability on number theory was mainly from his older
brother PAN Chengdong. Pan Chengdong was the graduate student of Prof. Min
(1956-1959). But Pan Chengdong was also considered to be the student of
Prof. Hua, especially on the research of Goldbach conjecture. Even Prof.
Min, was much influenced by Prof. Hua."
4 Around 1984, Prof. Shing-Tung Yau tried to arrange Zhang to go to UC San
Diego to study with the well-known analytic number theorist Harold Stark
there. Unfortunately, for some reasons this idea was vetoed. Otherwise he
might move academically along a path which is closer to a geodesic.
| i****g
发帖数: 3896 | 2 4 Around 1984, Prof. Shing-Tung Yau tried to arrange Zhang to go to UC San
Diego to study with the well-known analytic number theorist Harold Stark
there. Unfortunately, for some reasons this idea was vetoed. Otherwise he
might move academically along a path which is closer to a geodesic.
of
on
【在 i****g 的大作中提到】
: http://blog.sina.com.cn/s/blog_c24597bf0101b871.html
: 致谢：I would like to thank Prof. Shing-Tung Yau for suggesting the title of
:
: this article, Prof. William Dunham for information on the history of the
: Twin Prime Conjecture, Prof. Liming Ge for biographic information about
: Yitang Zhang, Prof. Shiu-Yuen Cheng for pointing out the paper of
: Soundararajan cited in this article, Prof. Lo Yang for information about
: Chengbiao Pan quoted below, and Prof. Yuan Wang for detailed information on
: results related to the twin prime conjecture, and Prof. John Coates for
: reading this article carefully and for several valuable suggestions.
|
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