 Physicists edge closer to the solution of ‘million dollar’ mathematical problem

Bernhard Riemann in 1863 (Credit: Public Domain)

Did a team of mathematicians to take a large step in the direction of the answer of a 160-year-old, million-dollar-question in mathematics?

Maybe. The crew did solve a number of other, smaller questions in a field called number theory. And in doing so, they have reopened an old road that would eventually lead to an answer to the question: Is the Riemann hypothesis correct?

The Reimann hypothesis is a fundamental mathematical conjecture that has huge implications for the rest of the math. It forms the basis for many other mathematical ideas, but no one knows if it is true. The validity has become one of the most famous open questions in mathematics. It is one of the seven “Millennium Problems” were created in 2000, with the promise that whoever solves them will win the \$1 million. (Just one of the problems is fixed now.) [5 Seriously Mind-Boggling Math Facts]

Where does this idea come from?

Back in 1859, a German mathematician called Bernhard Riemann presented an answer to a particularly tricky mathematical equation. His hypothesis goes as follows: The real part of any non-zero of the Riemann zeta function is 1/2. That is a fairly abstract mathematical statement, that have to do with what numbers you can place in a certain mathematical function to make that function equal to zero. But it turns out to be a great deal, especially with respect to questions of how often you will encounter prime numbers as you count up in the direction of the infinite.

We come back to the details of the hypothesis later. But the most important thing to know is, is that if the Riemann hypothesis is true, it answers a lot of questions in mathematics.

“So often, in number theory, what happens is if you assume that the Riemann hypothesis [is true], you are then able to prove many other results,” Lola Thompson, a number theorist at Oberlin College in Ohio, who was not involved in this latest research, said.

Often, she told Live Science, some theorists will first prove that something is true if the Riemann hypothesis is true. Then they will use that evidence as a kind of stepping stone to a more complicated proof, which shows that their original conclusion applies regardless of whether the Riemann hypothesis is true.

The fact that this trick works, ” she said, convinced many mathematicians that the Riemann hypothesis should be satisfied.

But the truth is that nobody knows for sure.

A small step in the direction of a proof?

So how did this small team of mathematicians seem to bring us closer in the direction of a solution?

“What we have done in our paper,” said Ken Ono, a number theorist at Emory University and co-author of the new evidence, “is that we revisited a very technical criterion equivalent to the Riemann hypothesis … and we have proven to be a big part of it. We have proven to be a big part of this criterion.”

A criterion equivalent to the Riemann hypothesis,” in this case refers to a separate declaration that is mathematically equivalent to the Riemann hypothesis.

It is not clear at first sight is the reason why the two statements are so connected with each other. (The criterion has to do with the so-called “hyperbolicity of Jensen polynomials.”) But in the 1920s, a Hungarian mathematician, George Pólya showed that when this criterion is met, then the Riemann hypothesis is true — and vice versa. It is an old proposed route in the direction of proving the hypothesis, but that was largely deserted.

Ono and his colleagues, in a paper published May 21 in the journal Proceedings of the Natural Academy of Sciences (PNAS), showed that in many cases the criterion is met.

But in mathematics, much is not enough to count as evidence. There are still a number of cases where they don’t know if the criterion is true or false.

“It’s like playing a million-number of Powerball,” Ono said. “And you already know the numbers, but the last 20. If even one of the last 20 numbers is wrong, you lose. … It could still all fall apart.”

Researchers would need to come up with a more advanced evidence of the criterion that applies in all cases, so the proof of the Riemann hypothesis. And it is not clear how far such evidence is, Ono said.

So, how big of a deal is this paper?

In terms of the Riemann hypothesis, it is difficult to say how important this is. A lot depends on what happens.

“This [the criterion] is just one of the many equivalent formulations of the Riemann hypothesis,” Thompson said.

In other words, there are a lot of other ideas, such as this criterion would prove that the Riemann hypothesis is true if they themselves were proven.

“So, it really is difficult to know how much progress this is, because on the one hand, that progress in this direction. But, there is so many equivalent formulations, perhaps, that this is the direction you do not intend to use the proceeds from the Riemann hypothesis. Maybe one of the other equivalent statements, but instead, if someone can prove that one of them,” Thompson said.

If the proof is along this track, then that will probably mean Ono and his colleagues have developed an important underlying framework for the resolution of the Riemann hypothesis. But if it is somewhere else, then this paper will prove to be less important.

Still, mathematicians are under the impression.

“Although this is still far away from the proof of the Riemann hypothesis, it is a big step forward,” Encrico Bombieri, a Princeton number theorist who was not involved in the team research, writes in an accompanying 23 May PNAS article. “There is no doubt that this paper will inspire further fundamental work in other areas of number theory and mathematical physics.”

(Bombieri was awarded a Fields Medal — the most prestigious prize in mathematics in 1974, in a large part for the work in connection with the Riemann hypothesis.)

What does the Riemann hypothesis mean anyway?

I promised we would return to this. Here is the Riemann hypothesis again: The real part of any non-zero of the Riemann zeta function is 1/2.

Let’s break that on the basis of the way in which Thompson and Ono explained.

First, what is the Riemann-zeta function?

In mathematics, a function is a relationship between the various mathematical quantities. Easy could look like this: y = 2x.

The Riemann zeta function follows the same principles. Only it is much more complicated. Here is how it looks like.

It is a sum of an infinite series, where each term — the first few are 1/1^s, 1/2^s and 1/3^s — is added to the previous terms and conditions. These ellipses mean that the series in the function remains but to go on that way forever.

Now we can answer the second question: What is a zero of the Riemann zeta-function?

This is easier. A “zero” of the function is a number you can put in for x that makes the function equal to zero.

Next question: What is the “real part of one of the zeros, and what does it mean that it is equal to 1/2?

The Riemann zeta function relates to what mathematicians call “complex numbers.” A complex number looks like this: a+b*i.

In this equation, “a” and “b” stand for all real numbers. A real number can vary from minus 3 to zero to 4.9234, pi, or 1 billion. But there is another kind of number: imaginary numbers. Imaginary numbers arise when you take the square root of a negative number, and they are important to you, displayed in all kinds of mathematical contexts. [10 Surprising Facts About Pi]

The simplest imaginary number is the square root of -1, which is written as “I.” A complex number is a real number (“a”) plus another number (“b”) times I. The real part of a complex number is that “one.”

A pair of zeros of the Riemann zeta-function, negative integers between -10 and 0, do not count for the Reimann hypothesis. These are considered to be “trivial” zeros because they are real numbers, not complex numbers. All other zeros are “non-trivial” and the complex numbers.

The Riemann hypothesis states that when the Riemann-zeta-function crosses zero (except those zeros between -10 and 0), the real part of the complex number is equal to 1/2.

That small claim that might not sound very important. But it is. And we may be just a teensy bit closer to the solution.

Originally published on Live Science.

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