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The real part of every non-trivial zero of the Riemann zeta function is 1/2
Lindelöf's hypothesis, one of the most important open problems in the history of mathematics, states that for large t, Riemann's zeta function ζ(1/2+it) is of order O(t^ε) for any ε>0. It is well known that for large t, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum.
“My approach was completely different from the usual approaches used,” Fokas said. “I first embed the R zeta function inside a bigger problem, namely I find that the R zeta function satisfies a very important problem in complex analysis called the Riemann-Hilbert problem. Then, I compute the large t behavior of this problem. Besides being conceptually novel, this approach is technically very hard due to the analysis of the aforementioned Riemann-Hilbert problem.”
-Athanassios Fokas, Mathematicia n-M.D. solves one of the greatest open problems in the history of mathematics (blog)
If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?
-David Hilbert, Wikiquote
Fokas is a world expert in asymptotics, an applied mathematical domain which helps scientists answer questions about the behavior of functions when a parameter is very large. His proof of Lindelöf also means a breakthrough in understanding algorithmic complexity, a very important topic in computer science. Knowing the complexity of algorithms allows us to answer questions such as how long will a program run on an input? How much space will it take? Is the problem solvable?
What can the equation be used for?
It's about a mathematical understanding of quantum field theories. These belong to the field of physics and play a part in large-scale experiments such as those carried out at CERN. The aim is to mathematically describe elementary particles, i.e. the smallest known components of matter. But this is so complicated that, instead, imaginary particles are described mathematically which have certain properties of the real particles. The hope is that one day the real particles can be described using the methods established in this way.
After working on the problem for 10 years, you experienced a breakthrough this year. How did that come about?
Towards the end of May, I tried out an idea for which my Ph.D. student, Alexander Hock, provided the decisive impulse. I worked out a new equation – simpler than the previous one – and began to solve it in loops. What this means is that you approach the solution step by step, i.e. loop by loop, by calculating the left side of the equation in each previous step and using it for the right side of the equation in the next step.
And now, there's a rumor circulating that this 160-year-old mega-problem might finally be solved – by British-Lebanese geometer Sir Michael Atiyah.
Intriguingly, by citing von Neumann, Hirzebruch and Dirac, Atiyah hints that his "simple proof" draws influence from the world of quantum mechanics. There's no doubt we should keep a healthy dose of skepticism until the proof is presented and reviewed. However, if it holds up, we could be about to see the most significant discovery in mathematics for over a quarter of a century.
The “boundary condition” is imposing an identification with zeta zeros by fiat, so the linkage of any of this to RH is basically circular. The paper at best just redefines the problem, without providing any genuine new insight. More specifically, as the experience of more than 100 years has shown, there are a zillion ways to recast RH without providing any real progress; this is yet another (if it makes any rigorous sense, which it does not yet do, yet the absence of rigor is not the reason for skepticism about the value of this paper, whatever the pedigree of the authors may be).
One has to find a way of encoding the zeta function that is not tautological (unlike the case here), and that is where deep input from number theory would have to come in. This is really the essential point that all papers of this sort fail to recognize.
Real insight into the structures surrounding RH have arisen over the past decades, such as the work of Grothendieck and Deligne in the function field analogue that provided a spectral interpretation through the development of striking new tools inspired by novel insights of Weil. In particular, the appearance of the appropriate zeta functions in such settings is not imposed by fiat, but is the outcome of a massive amount of highly non-trivial constructions and arguments. In another direction, compelling evidence and insight has come from the “random matrix theory” of the past couple of decades (work of Katz-Sarnak et al.) was inspired by observations originating with Dyson merged with work of the number theorist Montgomery.
Number theorists making a major advance on the puzzles of quantum gravity without providing an identifiable new physical insight is about as likely as physicists making a real advance towards RH without providing an identifiable new number-theoretic insight.
Abstract : The Riemann Hypothesis is a famous unsolved problem dating from 1859. I will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).
Sir Michael Atiyah, twitter
The competing mathematical strategies are manifest in a more recent discussion concerning the mathematical foundations of quantum mechanics. In the preface to von Neumann's treatise (1955) on that topic, he notes that Dirac provides a very elegant and powerful formal framework for quantum mechanics, but complains about the central role in that framework of an “improper function with self-contradictory properties,” which he also characterizes as a “mathematical fiction.” He is referring to the Dirac delta function, which has the following incompatible properties: it is defined over the real line, is zero everywhere except for one point at which it is infinite, and yields unity when integrated over the real line. Von Neumann promotes an alternative framework, which he characterizes as being “just as clear and unified, but without mathematical objections.” He emphasizes that his framework is not merely a refinement of Dirac's; rather, it is a radically different framework that is based on Hilbert's theory of operators.
Atiyah is well aware of this history of failure. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he says, but he hopes his presentation will convince his critics.
In it, he pays tribute to the work of two great 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, whose developments he claims laid the foundations for his own proposed proof. “It fell into my lap, I had to pick it up,” he says.
New Scientist contacted a number of mathematicians to comment on the claimed proof, but all of them declined.
The crux of Atiyah’s proof depends on a quantity in physics called the fine structure constant, which describes the strength and nature of electromagnetic interaction between charged particles. By describing this constant using a relatively obscure relationship known as the Todd function, Atiyah claimed to be able to prove the Riemann hypothesis by contradiction.
Atiyah’s proof claims to answer this question by relying on something he called the “Todd function,” named after the late mathematician and Atiyah’s former teacher J.A. Todd. As Pössel pointed out, the novelty of this function is the source of many mathematicians’ skepticism about Atiyah’s proof.
There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!
— Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 978-0-691-08388-9.
originally posted by: TEOTWAWKIAIFF
a reply to: Groot
People's eyes kind of roll back in their head when they see equations!! That is why editors tell science authors to leave equations out of their books. *face palms*
Seeing the math makes things easier (in my opinion). You can spend a paragraph of words and explaining instead of looking at the equation and saying, "Oh, it is an inverse relationship."
Glad to have pointed you over!
As Atiyah notes, in some sense α is a fundamental dimensionless numerical quantity like e or π. As such it is tempting to try to “derive” its value from some deeper principles. Arthur Eddington famously tried to derive exactly 1/137, but failed; Atiyah cites him approvingly.
But to a modern physicist, this seems like a misguided quest. First, because renormalization theory teaches us that α isn’t really a number at all; it’s a function. In particular, it’s a function of the total amount of momentum involved in the interaction you are considering. Essentially, the strength of electromagnetism is slightly different for processes happening at different energies. Atiyah isn’t even trying to derive a function, just a number.
Point to Point
If you’re handed an equation, you can graph its solutions and produce a curve. Mathematicians want to know how many of these solutions are rational numbers — values that can be expressed as a ratio of two integers (such as 1/2, −3, or 4483/929).
Rational solutions are hard to find systematically, but mathematicians have techniques that work under some circumstances. Say you have the equation x^2 + y^2 = 1. The graph of the solutions to this equation form a circle. To find all the rational points on that circle, start with one particular rational solution — say, the point on the circle where x is 1 and y is 0. Then draw a line through that point that intersects the circle at one other point. So long as the slope of your line is rational, the second point of intersection will also be a rational solution. Through the line-drawing procedure, you’ve parlayed one rational solution into two.