Scientists in the UK, US and Canada have made a significant breakthrough in attempting to establish the Hilbert-Pólya conjecture.
A study published on 31 March 2017 by Brunel University London, Washington University St. Louis and University of Western Ontario gives fresh hope that it might be possible to prove the validity of the Riemann hypothesis - the Maths world's Holy Grail - based on what is known as the Hilbert-Pólya construction.
First posed by Bernhard Riemann in 1859 but not yet proven true or false, the Riemann hypothesis asserts that the points at which the Riemann zeta function vanishes lie on a special straight line. The function is useful in number theory, such as for investigating properties of prime numbers.
Yet in the century and a half since, and in spite of hard efforts by many mathematicians, no one has been able to prove that all of the (nontrivial) zeros lie on that critical line.
Solving this grand challenge would have an immense impact on many branches of mathematics. It is also one of a handful of major maths problems to carry a million dollar reward.
Because the zeros of the zeta function appear as discrete points, one promising way of proving the hypothesis – known as the Hilbert-Pólya programme – is to find an operator whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function, and then to prove that these eigenvalues are real numbers.
The Riemann zeta function operates in the field of complex numbers, which combine real numbers (the type of numbers we’re all familiar with) with imaginary numbers (related to the square root of –1), so it is a special case if the eigenvalues have no imaginary part. Attempts have been made for many decades to find such an operator, with no success until now.
Writing in the journal Physical Review Letters, Professor Dorje Brody (Brunel University London), Professor Carl Bender (Washington University St. Louis) and Dr Markus Müller (University of Western Ontario) explain how such an operator can be obtained by explicitly constructing one. Thus, the veracity of the Riemann hypothesis can be established by showing the reality of the eigenvalues of a relatively simple operator they have discovered.
Although they have not proved the reality of the eigenvalues, the researchers present a heuristic analysis – a close but imperfect test – strongly suggesting that this is indeed the case.
The secret behind their insight is to borrow ideas from a recent development in physics – known as PT-symmetric quantum theory – in which physical systems are described not by conventional Hermitian operators guaranteed to possess real eigenvalues, but rather by pseudo-Hermitian operators that are invariant under space-time reflection.
Such operators can, but not necessarily, have entirely real eigenvalues, and are known to fulfil certain properties. The researchers show that these properties are satisfied only if the Riemann hypothesis is true. If the analysis presented can be made rigorous, then it would constitute proof of the Riemann hypothesis.
Professor Brody concludes: “While we do not claim proof of the Riemann hypothesis, our study provides ample heuristic results that are highly suggestive that the hypothesis is a valid one, and we should remain optimistic that one day we will be able to confirm this.”
(Image: Bernhard Riemann in 1863)
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