The hot news this week from the mathematical physics world is that the noted mathematician Michael Atiyah claimed to have solved the Riemann hypothesis, one of the most difficult unsolved problems known and whose resolution carries a $1 million prize. The problem is that Atiyah’s solution, while remarkable for its brevity, may not hold water.
The Riemann hypothesis is concerned with the Riemann zeta function, which – in very broad terms – provides a way to predict the position of prime numbers on the number line. Computers have been able to find prime numbers with scores of digits and mathematicians have been able to find in hindsight that, yes, the zeta function predicts they exist. However, what mathematicians don’t know (and this is the Riemann hypothesis) is whether the function can predict prime numbers ad infinitum or if it will break at some particularly large value. And solving the Riemann hypothesis problem means proving that the zeta function can indeed predict the position of all prime numbers on the number line.
A more technical explanation, reproduced from my article in The Wire last year, follows; article continues below this section:
In 1859, Bernhard Riemann expanded on Euler’s work to develop a mathematical function that relates the behaviour of positive integers, prime numbers and imaginary numbers. The Riemann hypothesis is founded on a function called the Riemann zeta function. Before him, Euler had formulated a mathematical series called Z (s), such that:
Z (s) = (1/1s) + (1/2s) + (1/3s) + (1/4s) + …
He found that Z (2) – i.e., substituting 2 for s in the Z function – equalled π2/6, and Z (4) equalled π4/90. At the same time, for many other values of s, the series Z (s) would not converge at a finite value: the value of each term would keep building to larger and larger numbers, unto infinity. This was particularly true for all values of s less than or equal to 1.
Euler was also able to find a prime number connection. Though the denominators together constituted the series of positive integers, with a small tweak, Z (s) could be expressed using prime numbers alone as well:
Z (s) = [1/(1 – 1/2s)] * [1/(1 – 1/3s)] * [1/(1 – 1/5s)] * [1/(1 – 1/7s)] * …
This was Euler’s last contribution to the topic. In the late 1850s, Riemann picked up where Euler left off. And he was bothered by the behaviour of the series of additions in Z (s) when the value of s dropped below 1.
In an attempt to make it less awkward (nobody likes infinities), he tried to modify it such that Z (2) and Z (4), etc., would still converge to interesting values like π2/6 and π4/90, etc. – but while Z (s ≤ 1) wouldn’t run away towards infinity. He succeeded in finding such a function but it was far more complex than Z (s). This function is called the Riemann zeta (ζ) function: ζ (s). And it has some weird properties of its own.
One such is involved in the Riemann hypothesis. Riemann found that ζ (s) would equal zero whenever s was a negative even number (-2, -4, -6, etc.). These values are also called trivial zeroes. He wanted to know which other values of s would precipitate a ζ (s) equalling zero – i.e. the non-trivial zeroes. And he did find some values. They all had something in common because they looked like this: (1/2) + 14.134725142i, (1/2) + 21.022039639i, (1/2) + 25.010857580i, etc. (i is the imaginary number represented as the square-root of -1.)
Obviously, Riemann was prompted to ask another question – the question that has since been found to be extremely difficult to answer, a question worth $1 million. He asked: Do all values of s that are non-negative integers and for which ζ (s) = 0 take the form ‘(1/2) + a real number multiplied by i‘?
In more mathematical terms: “The Riemann hypothesis states that the nontrivial zeros of ζ (s) lie on the line Re (s) = 1/2.”
When I first heard Atiyah’s claim, I was at a loss for how to react. Most claimed solutions for the Riemann hypothesis are usually dismissed quickly because they contain leaps of logic not backed by sufficient mathematical rigour. On the other hand, Atiyah isn’t just anybody. He won the Fields Medal in 1966 and the Abel Prize in 2004, and has been associated with some famous solutions for problems in algebraic topology.
Perhaps the most famous and recent example of this was Vinay Deolalikar’s proof of another major unsolved problem in mathematics, whether P equals NP, in August 2010. The P/NP problem asks whether a problem whose solution is easy to check is also therefore easy to solve. Though nobody has been able to provide a proof for this conundrum yet, it is widely assumed by mathematicians and computer scientists that P = NP, i.e. a problem whose solution is easy to check is therefore also easy to solve. However, Deolalikar, then working at Hewlett Packard Research Labs, claimed to have a proof that P ≠ NP, and it couldn’t be readily dismissed either because, to borrow Scott Aaronson’s words,
What’s obvious from even a superficial reading is that Deolalikar’s manuscript is well-written, and that it discusses the history, background, and difficulties of the P vs. NP question in a competent way. More importantly (and in contrast to 98% of claimed P≠NP proofs), even if this attempt fails, it seems to introduce some thought-provoking new ideas, particularly a connection between statistical physics and the first-order logic characterization of NP.
Nonetheless, flaws were found in Deolalikar’s proof, as delineated prominently in Aaronson’s and R.J. Lipton’s blogs, and the claim was settled: P/NP remained (and remains) unsolved. Lesson: watch the blogs as a first response measure. The peers of a paper’s author(s) usually know what’s happening before the news does and, if a controversial claim has been advanced, they’re likely already further into a debate than the mainstream media realises.
So as a quick way out in Atiyah’s case, I hopped over to Shtetl Optimized, Aaronson’s blog. And there, at the end of a long post about the weirdness of quantum theory, was this line: “As of Sept. 25, 2018, it is the official editorial stance of Shtetl-Optimized that the Riemann Hypothesis and the abc conjecture both remain open problems.” Aha!
Some of you will remember that three physicists made a major announcement last year about finding a potential way to solve the Riemann hypothesis because they had unearthed an eerie similarity between the Riemann zeta function, central to the hypothesis, and an equation found in quantum mechanics. While they’re yet to post an update, the physicists’ thesis was compelling and wasn’t dismissed by the wider mathematical community, raising hope that it could lead to a solution.
Atiyah’s solution also concerns itself with a famously physical concept: the fine-structure constant, denoted as α (alpha). The value of this constant determines the strength with which charged particles like electrons interact with the electromagnetic field. It has the value of about 1/137. If it were higher, the electromagnetic force would be stronger and all atoms would be smaller, apart from numerous other cascading effects. Atiyah’s resolution of the Riemann hypothesis is pegged to a new derivation for the value of α, and this where he runs into trouble.
Sean Carroll, a theoretical physicist Caltech, called the derivation “misguided”. Madhusudhan Raman, a postdoc at the Tata Institute of Fundamental Research, said that while he isn’t qualified to comment on the correctness on the Riemann hypothesis proof, he – like Carroll – had some problems with the physics of it.
His full explanation is as follows (paraphrased): It is tempting to think of α as a fixed number, like π (pi), but it is not. While the value of π does not change, the value of α does because it is related to the energy at which it is being measured. At higher energies, such as inside the Large Hadron Collider, the value of α will be higher. So α is not a number as much as a function that says its value is X at energy Y. However, Atiyah appears to have worked with the assumption that α is a single, fixed number like π. This isn’t true and therefore his derivation is suspect.
Sabine Hossenfelder, a research fellow at the Frankfurt Institute for Advanced Studies, also had the same issues with Atiyah’s effort. Carroll went a step further and said that if he had to be very charitable, then the derivation could pass muster but not without also discussing various issues in physics associated with α. However, he wrote, “Not a whit of this appears in Atiyah’s paper.”
At the same time – and unlike in numerous previous instances – these physicists and others besides continue to have great respect for Atiyah and his work, and why not? Though he is 89, as one comment observed on Carroll’s blog, “It’s brave to fight to the last, and, who knows, with his distinguished record and doubtless vast erudition, maybe there’s some truth or useful insights in these latest papers, even if [it’s] not quite what he claims.”
So also, the Riemann hypothesis endures unresolved.
September 28, 2018