Erdös-Turan conjecture: Green-Tao theorem solves a special case

Last week, on Christmas day to be precise, I happen to browse through a piece of article on the arithmetic progression of prime numbers. The full article [1] is interlaced below for reference. It is indeed a very interesting one concerning prime numbers. Loosely speaking, this brings out some curious structure (perhaps I cant quite strictly say so because the word structure means much more than that), in the form of the a natural progression among prime numbers.

Let us get ourselves convinced by looking into an example,
199,
409,
619,
829,
1039,
1249,
1459,
1669,
1879,
2089

what is so noteworthy among these weird looking numbers? They are all prime numbers alright. What is more graceful is that they form an arithmetic progression with a difference of 210! We could call this set of prime numbers as a a 10-term arithmetic progression of primes with difference 210. Now this rather surprising example is not an isolated one in the realm of prime numbers. It has been long conjuncture'd that arbitrarily long sequence of prime numbers do exist among arithmetic progressions (of natural numbers). A proof eluded us for a long time.
Thankfully, in 2004, a formal claim came out to prove that, the prime numbers do contain arithmetic progressions of any arbitrary length (In the above example 10 was the length) ! This proof was proposed by mathematicians Ben Green and Terence Tao. They used an important result known as Szemeredis theorem in combination with work by Goldston and Yildirim, a clever "transference principle," to successfully establish the fundamental theorem that the prime numbers do contain arithmetic progressions of arbitrary length.

This amazing fact about prime numbers, now with a formal proof to its credit itself can be considered as a special case of another conjuncture known as Erdos-Turan conjuncture. Paul Erdos is someone you would instantly associate things in number theory to. So, this name probably doesn't surprise us. Anyway,

The Erdős-Turán conjecture is an unproven proposition in additive number theory. The conjecture states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

Mathematically speaking it states that, if

then A contains arithmetic progressions of any given length. If true, the theorem would generalize Szemerdis theorem. See [4] for a quick statement of Szemerdis theorem itself.

The article published in the newspaper The Hindu [1] is attached (inline) below:

Major progress in prime number theory

Krishnaswami Alladi

The Green-Tao theorem resolves an important special case of the Erdös-Turan conjecture

Kumbakonam: Professor Terence Tao of the University of California, Los Angeles (UCLA), was awarded the 2006 SASTRA's Ramanujan Prize at the International Conference on Number Theory and Combinatorics at the Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam.

This $10,000 prize comes on the heels of the Fields Medal that was awarded to Professor Tao in August for revolutionary contributions to several areas of mathematics.

Following the award ceremony on Ramanujan's birthday at Kumbakonam, Professor Tao delivered the Ramanujan Commemoration Lecture entitled "Long arithmetic progressions of primes," in which he reported major progress in prime number theory based on his recent work with Professor Ben Green of Cambridge University.

One of the most famous unsolved problems in mathematics is the Prime Twins Conjecture, which asserts that there are infinitely many prime pairs that differ by 2. More generally, the prime k-tuples conjecture states that if a k-tuple is admissible, then there are infinitely many such k-tuples of primes. Here by admissible one means that the k-tuple must satisfy certain non-divisibility conditions.

If the prime k-tuples conjecture is true, then it follows that there are arbitrarily long arithmetic progressions of primes. For example, 7, 37, 67, 97, 127, 157, is an arithmetic progression of 6 primes with common difference 30.

Sieve theory was developed in the 20th century to attack problems such as the k-tuples conjecture. Although this conjecture is still unsolved, sieve methods have succeeded in establishing similar results for almost primes, namely, those integers with very few prime factors, but not for the primes themselves.

Thus, the world was astonished when Professor Tao and Professor Green proved in 2003 that there are arbitrarily long arithmetic progressions of primes. The road to the Green-Tao theorem has been long, and in his lecture, Professor Tao surveyed the history of the problem and described the techniques that led to the recent breakthrough.

The first major advance was made in 1939 by van der Corput, who showed that there are infinitely many triples of primes in arithmetic progression. He used the circle method, originally invented by Hardy and Ramanujan to estimate the number of partitions of an integer and subsequently improved by Hardy and Littlewood to apply to a wide class of problems in additive number theory.

van der Corput's result was improved in 1981 by the British mathematician Heath Brown, who showed that there are infinitely many quadruples in arithmetic progression of which three are primes, and the fourth an almost prime with at most two prime factors. That such an improvement came after more than 40 years indicates the difficulty of the problem.

Another problem was the study of finite arithmetic progressions within sets of positive density. This was pioneered by the 1958 Fields medallist K.F. Roth, who in 1956 showed that any set of integers with positive density contains infinitely many triples in arithmetic progression. This study culminated in 1975 with the grand result of the Hungarian mathematician Szemeredi, who proved that any set of integers with positive density contains arithmetic progressions of arbitrary length. Professor Tim Gowers of Cambridge University, who won the Fields Medal in 1994, has recently given a simpler proof of Szemeredi's theorem. It is to be noted that since the primes have zero density, Szemeredi's theorem does not imply that there are arbitrarily long arithmetic progressions of primes.

Professor Green was a Ph.D student of Professor Gowers, who introduced him to Szemeredi's theorem. One of Professor Green's first major accomplishments was the result that any subset of the primes, which has relative positive density, contains infinitely many triples on arithmetic progressions. Professor Tao and Professor Green then corresponded due to their common interest on such problems. They studied the general problem of arithmetic progressions in sparse sets of integers. By combining ideas from ergodic theory, the techniques of Professor Gowers, and repeated use of Szemeredi's theorem, they were able to prove the astonishing result that there are arbitrarily long arithmetic progressions of primes. The ingredients of the proof were put together when Professor Green visited Professor Tao at UCLA in 2003.

The great Hungarian mathematicians Paul Erdös and Paul Turan conjectured that if A is an infinite set of integers the sum of whose reciprocals is divergent, then there are arbitrarily long arithmetic progressions in A. Since the sum of the reciprocals of the primes is a divergent series, the Green-Tao theorem is a special case of the Erdös-Turan conjecture, which remains unsolved in full generality. Erdös has offered $10,000 for a resolution of this conjecture. The Green-Tao theorem resolves an important special case of the Erdös-Turan conjecture and is a phenomenal achievement by two brilliant young mathematicians. Thus, it was a fitting tribute to Ramanujan that this great work was presented in his hometown on his birthday.

References:
[1]http://www.hindu.com/2006/12/25/stories/2006122502481100.htm
[2]Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions,8 Apr 2004.
[3] Terence Tao, Tamar Ziegler, The primes contain arbitrarily long polynomial progressions
[4] http://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem