Mathematics in its prime!

Work recently completed by Dr. Tammy Ziegler of the Department of Mathematics together with colleagues from the U.S. and Britain presents a major advance in Number Theory

Dr. Tammy Ziegler

By Noam Bercovitz

The conversation with Dr. Tammy Ziegler of the Department of Mathematics took place, by pure chance, on the same day that it was announced that Prof. Elon Lindenstrauss of the Hebrew University was the recipient of the Fields Medal – the most prestigious mathematics prize in the world, which is, for mathematicians, the equivalent of the Nobel Prize. Dr. Ziegler, who got her degrees at the Hebrew University, knows Prof. Lindenstrauss and his work well, and was excited and happy for him and the mathematics community in Israel. This feeling  was also reflected in the Israeli media, which proudly reported the achievement, while being somewhat surprised to note that high-level research in mathematics was actually taking place, and that Israel, as it became clear, was a major player in the field.

Dr. Ziegler has been in the Technion’s Department of Mathematics for three years. Work that she and her two colleagues, Professor Tao from UCLA and Professor Green of Cambridge University, have recently completed, has aroused much interest among mathematicians since it solves basic problems in the field of prime numbers – a mathematical field that has lately become a center of attention, after a period of slumber. The results delineate methods for finding asymptotics for arithmetic patterns of prime numbers. The solution combines methods from two seemingly unrelated fields – dynamics and number theory.

Two thousand years

The fascination with prime numbers, explains Dr. Ziegler, is almost as old as mathematics. Already about 2,300 years ago, Euclid showed that each natural number (except for 1) can be written as a unique product of prime numbers. Furthermore, Euclid proved that there is an infinite number of prime numbers. His reductio ad absurdum proof is still considered to be one of the most elegant mathematical proofs. It states: let us assume that there is a finite number of prime numbers that can be written entirely as the sequence P₁< P₂<…<P_k. Now let us look at the number that is the product of all the elements in the series plus 1:

M = P₁*P₂*...*P_k + 1

Here M is larger than P_k and is, therefore, not prime, but M is also not divisible by any element of the sequence It will always have a remainder of 1. This contradicts the assumption that there is a finite number of prime numbers.

From this, according to Ziegler, comes the question of how frequently do prime numbers appear? A quantitative estimate would be very nice to have. We know intuitively that there are more even number than numbers divisible by three, and more numbers divisible by three than numbers that are perfect square roots. Indeed, If we take a very large number (say, N =10⁹), we know that it has about  N/2 even numbers, about N/3 numbers that are divisible by 3, and about  perfect square roots. Dr. Ziegler explains that in these cases, the estimation is easy; in contrast, Euclid’s proof  does not provide a way for  estimating the number of prime numbers smaller than N.

More than 2000 years passed before a formula, stating that there are about N/lnN prime numbers smaller than N, was established. The formula was conjectured by Gauss and Legendre, based on numerical data, and proved independently by Hadamard and de la Vallée Poussin in 1896.

The problem of prime number pairs (twins)

The next step, according to Ziegler, involved finding arithmetic patterns in the sequence of prime numbers. The question is interesting because of the inherent difficulty in understanding the additive behavior of prime numbers. For example, many prime number pairs that differ from each other by two (called "twin primes") are known. It is tempting to conjecture that there might be an infinite number of such pairs, but to-date the answer to this question has eluded mathematicians; it remains an open problem.

A related question that has interested mathematicians concerns the existence of arithmetic progressions in the sequence of primes. Only in 2004 did Green and Tao achieve a breakthrough by showing that the set of prime numbers contains arbitrarily long arithmetic progressions. The two researchers approached the problem from a different and surprising direction, using ideas from Ergodic theory, which is a branch of mathematics that deals with the study of dynamic systems. Green and Tao proved the existence of arithmetic progressions  of prime numbers, but their methods did not provide estimates of the number of k-term arithmetic progressions of prime numbers, all of  whose elements are smaller than N.

Dr. Ziegler’s doctorate focused on the connection between the arithmetic progressions and nilpotent dynamic systems. Prof .Tao's interest in this work led to their collaboration. About three years ago Ziegler started working with both Green and Tao and the collaboration resulted in finding the important estimates that have aroused so much interest.

When things get complicated

Trying to explain her unique contribution to the solution of the problem, Dr. Ziegler finds it is too complicated to do in simple terms. The explanation would involve sophisticated concepts and require the reader to possess advanced knowledge of Mathematics; thus, it is beyond the scope of this interview.

In mathematical language, one of the conclusions of the work of Green, Tao and Ziegler is that each system of equations of finite complexity, or in other words, a system that is not hiding within it a problem similar to prime twins, has prime solutions unless there are “local obstructions”, and thereby corroborates a multidimensional generalization for Hardy and Littlewood’s conjectures of the early twentieth century.

With paper and pencil

It is intriguing to find out how mathematicians work, and Dr. Ziegler explains with a smile: a lot of work and not being afraid to try new ideas. Dr. Ziegler relates that her office in the department provides a pleasant and quiet environment, and sometimes in the evenings she goes to a café with a notepad and a pencil.

Her work involves thinking hard, discussions with colleagues both here and overseas in order to analyze the problem and come up with new ideas. Finally, once you get an idea for a solution, you have to try and write it out in full detail. In most cases, though, you reach a dead end, which means that a significant part of your work ends up with tossing away ideas that at first looked promising. There is also no guarantee at the start of the road that a solution will be found at its end, therefore, when you do reach a solution, such as Ziegler and her colleagues did, there is a sense of accomplishment.