Retired mathematician Sir Michael Atiyah claims to have demonstrated a simple solution to the Riemann hypothesis, which has remained unsolved for 159 years. He is set to formally announce his explanation– which could have massive implications for digital security — on Monday.
Over the centuries, mathematicians have approached more and more complex problems. Some problems have remained stubbornly opaque, defying decades and even centuries of attempts to solve them. The most famous (and most exciting ones) are probably the so-called Millennium Prize Problems: seven problems stated by the Clay Mathematics Institute back in 2000, the solution to which will grant the author $1 million — along with ever-lasting fame and respect, of course. Only one has been solved so far, but a senior mathematician claims to have found the solution to another one.
The mathematician in case is Michael Atiyah, and he’s probably not the kind of person you’d expect to solve a Millennium Problem; not because he doesn’t have the capacity to do that — you could hardly find a more prolific and respected mathematician today — but because at 90 years old, he’s seemingly long retired.
Atiyah is aware of the long history of failures surrounding the Riemann hypothesis:
“Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he says, but he is still confident in his proof. Atiyah, a laureate of the two most prestigious awards in mathematics (the Fields Medal and the Abel Prize) is set to present his recent work at the Heidelberg Laureate Forum, at an event involving some of the world’s brightest mathematicians.
While no other mathematicians have officially commented on this, Twitter reactions from other mathematicians have been mixed, ranging from excited to very skeptical. This claim is not to be taken lightly, however: Atiyah is renowned and very well respected, and to say that he is credible is an understatement. To make matters even more intriguing, Atiyah claims that his proof is “simple” — though in this context, “simple” means “relatively simple.”
The implications for this goes way beyond a theoretical math problem — much of today’s cryptography and digital security relies on this random distribution. Essentially, all modern cryptography relies on the fact that prime numbers occur sporadically. Atiyah’s solution could raise new challenges for this approach, potentially bringing a paradigm shift in modern cryptography.
All the more reason to keep an eye out for his announcement.
The Riemann hypothesis
The Riemann hypothesis starts with prime numbers — rather strange numbers which can’t be divided by other numbers; one of the oddities of prime numbers is that their distribution is irregular — there’s no precise method to predict where the next prime number will occur. This unpredictability has been widely used in developing digital security systems.
While looking at prime numbers, Berhard Reimann, one of the most prolific German mathematicians, realized something interesting: the distribution of these prime numbers isn’t random at all, it’s very similar to a function, called the Riemann Zeta Function, described below.
ζ(s) = 1/1s + 1/2s + 1/3s + 1/4s + …. up to infinity
This is where it starts to get tricky. The variable s can take any value, and Riemann’s work gives us an explicit formula for the number of primes in a given interval. This is done in terms of the so-called ‘zeros’ of the function — the values of s under which the function becomes 0. You can think of this function as a way to predict the distribution of prime numbers.
Riemann observed this in action, but he was never really able to prove it. This is why this is still a hypothesis, and not a theory (which, in science and math, has a much stricter meaning than in regular talk). It remains to be seen whether this will still be the case after Monday or not.
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