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New Intelligence Report

Editor: Aeneas is so sleepy

【New Wisdom Introduction】MIT mathematics professor Larry Guth and Oxford University Fields Medal winner James Maynard made a major breakthrough in the Riemann hypothesis, directly breaking the record of more than 80 years. Interestingly, in the process they sacrificed a "discarded piece", making the situation more complicated and difficult, but closer to the answer.

A significant breakthrough has been made in the Riemann hypothesis (RH), one of the "Seven Great Mathematical Problems of the Millennium", and mathematicians are one step closer to winning the "crown of the conjecture world"!

MIT has proposed stricter restrictions on potential exceptions to the Riemann hypothesis, directly breaking a record that had stood for more than 80 years.

Paper address: https://arxiv.org/abs/2405.20552

Today, the Riemann hypothesis remains one of the most important unsolved mysteries in mathematics. If it can be proved, mathematicians will have a deeper understanding of the distribution of prime numbers.

Moreover, a lot of work in the field of number theory and complex functions is based on the premise that the Riemann hypothesis is true, so once the Riemann hypothesis is proved, many other works will also be fully proved.

Whoever solves the Riemann hypothesis will receive a $1 million reward from the Clay Mathematics Institute.

Mathematicians currently have no idea how to prove the Riemann hypothesis, but they can still obtain useful results by proving that the number of possible exceptions is finite.

In May, Maynard and MIT’s Larry Guth established a new cap on the number of a particular type of exception, breaking a record that had stood for more than 80 years.

With their new proof, they get a better approximation for the number of primes in short intervals on the number line and hope to provide more insights into the history of prime numbers.

Still a long way from fully solving the Riemann hypothesis, but a historic moment nonetheless

Henryk Iwaniec of Rutgers University commented: "This is a sensational result. This process was very, very difficult, but they pulled out a gem."

Terence Tao praised the paper highly:

Guth and Maynard made a remarkable breakthrough in the Riemann hypothesis by providing the first substantial improvement on the classical 1940 Ingham bounds on the zeros of the Riemann zeta function (and more generally, on the control of large values ​​of various Dirichlet series).

He considered this a historic moment, saying that "in the eighty years since the Riemann hypothesis existed, the only push on this constraint has been a tiny improvement in the error of (1)".

Although he also admitted that "we are still far from completely solving this conjecture."

You know, as early as 2008, Xian-Jin Li, a mathematician from Brigham Young University in the United States, also published a paper on arxiv, claiming to have proved the Riemann hypothesis. Later, Terence Tao and French mathematician Alain Connes (both Fields Medal winners) ruthlessly pointed out the errors in Li's proof.

Well, the fact that Guth and Maynard's research was forwarded by Terence Tao shows that it is of great significance.

A clever detour

The Riemann hypothesis involves a core formula in number theory - the Riemann zeta function. The zeta function is a generalized form of a simple summation:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯

This series will become infinite as the number of terms increases, a process mathematicians call "divergence". But if we sum it instead:

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ⋯ = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ⋯

We would get π^2/6, which is approximately 1.64.

Riemann made an unexpectedly great idea to turn such a series into a function as shown below:

So ζ(1) is infinite, but ζ(2) = π^2/6.

Things get really interesting when we make s a complex number.

Complex numbers have two parts: the real part, which is the number we see in everyday life, and the imaginary part, which is the everyday number multiplied by the square root of -1 (mathematicians call this i).

Complex numbers can be plotted on a plane with the real part on the x-axis and the imaginary part on the y-axis. For example, 3 + 4i.

The zeta function takes as input points on the complex plane and outputs other complex numbers.

It turns out that for certain complex numbers, the zeta function is zero. Pinning down where these zeros are in the complex plane is one of the most interesting problems in mathematics.

In 1859, Riemann conjectured that all zeros are concentrated on two lines. If we extend the zeta function to handle negative inputs, we will find that the zeta function is zero for all negative even numbers: -2, -4, -6, etc.

This is relatively easy to prove, and so these are called trivial zeros.

When the real part of s is less than 1, the sum of the entire series may diverge. In order to make the function applicable to a wider range, Riemann rewrote the above zeta function into the above form

Riemann conjectured that all other zeros of the function (i.e., nontrivial zeros) have real parts equal to 1/2 and therefore lie on this vertical line.

This is the Riemann hypothesis, and proving it has always been extremely difficult.

Mathematicians knew that the real part of every nontrivial zero must be between zero and 1, but they could not rule out the possibility that some zeros might have a real part of 0.499.

What they can do is show that such zeros must be extremely rare.

More intuitively, the zeta function can draw an infinite number of points. Riemann speculated that these points have a certain arrangement pattern, some are on a horizontal line, and the other part is on a vertical line. All these points are arranged on these two straight lines without exception.

In the above picture, since there are infinite points, we cannot use enumeration to prove that all the points are on these two lines, because we can never verify them all. But as long as there is one point that is not on these two lines, the Riemann hypothesis can be overturned.

Mathematicians have used computers to verify that the initial 100 trillion points all conform to the arrangement rules of the Riemann hypothesis.

Over the years, many mathematicians have tried to prove this conjecture, but no one has been able to bring back this "holy grail of mathematics". Some mathematicians even died with regret for this, leaving endless thoughts to future generations.

American mathematician Hugh Montgomery even said that if a devil agreed to let mathematicians exchange their souls for the proof of a mathematical proposition, most mathematicians would want to exchange it for the proof of the Riemann hypothesis.

The record of more than 80 years was suddenly broken

In 1940, a British mathematician named Albert Ingham established an upper limit for estimating the number of zeros whose real part is not equal to 1/2, and this upper limit is still used as a reference point by mathematicians today.

Decades later, in the 1960s and 1970s, other mathematicians found ways to transform Ingham’s results into descriptions of how prime numbers cluster or scatter on a number line, as well as other patterns they might form.

Around the same time, mathematicians also introduced new techniques that improved Ingham's upper limit on zeros with real parts greater than 3/4.

But it turns out that the most important zeros are those whose real part is exactly 3/4.

“Many important results about prime numbers are constrained by our understanding of the zeros whose real part is 3/4,” Maynard said.

James Maynard is an outstanding scholar in the field of mathematics and won the Fields Medal in 2022.

He received his undergraduate degree from the University of Cambridge and his PhD from the University of Oxford. He has been teaching at the Mathematical Institute of the University of Oxford since 2018.

About a decade ago, Maynard began thinking about how to improve Ingham's estimates of these specific zeros. "This is one of my favorite problems in analytic number theory. I always feel that if I work a little harder, I can make progress."

But year after year, whenever he tried to solve the problem, he was always stuck.

Then, in early 2020, on a plane to a conference in Colorado, an idea struck him — perhaps the tools from harmonic analysis could be useful.

Coincidentally, Larry Guth, an expert in harmonic analysis at MIT, happened to be attending the same meeting.

Two people who happened to be thinking about similar problems met each other.

However, Guth was completely unfamiliar with analytic number theory, so during lunch, Maynard explained the subject to him and gave him a concrete test case.

After several years of intermittent research, Guth realized that his harmonic analysis technique did not work.

But he didn't stop thinking about the problem and tried new approaches.

In February of this year, he contacted Maynard again, and the two began to work together seriously, combining their different perspectives.

A few months later, they had the results.

The "discarded pieces" in mathematics

Guth and Maynard began by translating the problem they wanted to solve into another form.

If the real part of some zero is not 1/2, then the correlation function, called the Dirichlet polynomial, must produce a very large value.

Therefore, proving that exceptions to the Riemann hypothesis are rare is equivalent to proving that the Dirichlet polynomials do not often produce very large values.

Then the mathematicians performed another transformation.

First, they built a matrix, or table of numbers, using the Dirichlet polynomials.

“Mathematicians love to see matrices because they’re something we know very well,” Guth said. “You learn to keep your nose open and be prepared to see matrices everywhere.”

A matrix can "act on" a mathematical object called a vector, which is defined by a length and a direction, to produce another vector.

Normally, when a matrix is ​​applied to a vector, it changes both the length and direction of the vector.

Sometimes there are special vectors that change only their length but not their direction when they pass through the matrix. These vectors are called eigenvectors.

Mathematicians measure the size of these changes using numbers called eigenvalues.

Guth and Maynard reformulated their problem so that it becomes about the largest eigenvalue of a matrix.

If they could prove that the largest eigenvalue couldn't get too large, their work would be done.

To do this, they used a formula that took a complex sum and looked for ways to make the positive and negative values ​​in the sum cancel each other out as much as possible.

“You have to rearrange the sequence or look at it from the right angle to see some kind of symmetry to achieve some cancellation,” Guth said.

The process involved several surprising steps, including one overarching idea that Maynard describes as “a little magical.”

At some point, they should have taken the seemingly obvious step of simplifying their sum.

However, they did not do this. Instead, they kept the sum in a longer, more complex form.

“We did something that at first glance seemed completely stupid, where we just refused to do the standard simplifications,” Maynard said. “We gave up a lot, which means now we can’t get any simple bounds for this sum.”

But in the long run, it proved to be a beneficial move.

“In chess, this is called a sacrifice—sacrificing a piece to gain a better position on the board,” Maynard said.

Guth likens it to playing with a Rubik's Cube: Sometimes you have to undo your previous moves, making everything look worse, and then find a way to get more colors to the right places.

“It takes a lot of courage to throw away an obvious improvement and then hope that you can recover it later,” said Roger Heath-Brown, a mathematician at the University of Oxford and Maynard’s former advisor. “It goes against everything I think should be done.”

But the mentor admitted that this was exactly where he was stuck.

Maynard said Guth's background as a harmonic analyst rather than a number theorist made this strategy possible. "He wasn't constrained by these rules, so he was more willing to consider things that didn't fit the rules."

Eventually, they were able to place a good enough bound on the largest eigenvalue, which in turn translated into a more precise bound on the number of potential counterexamples to the Riemann hypothesis.

Although their work began with the ideas of harmonic analysis that inspired Guth, they ultimately excluded these complex techniques and returned to basics.

“Now it seems like something I might have tried 40 years ago,” Heath-Brown said.

Finally, by giving better bounds on the number of zeros with real part 3/4, Guth and Maynard automatically proved some results about the distribution of prime numbers.

For example, estimates of the number of primes found in a given interval become less accurate for shorter intervals, and the new work could allow mathematicians to get good estimates for shorter intervals.

Mathematicians believe that this proof may also improve other conclusions about prime numbers.

And it seems that there is room for further improvement in Guth and Maynard's technique.

But Maynard believes that these techniques are not the right way to solve the Riemann hypothesis itself.

“It will also require some big ideas from elsewhere.”

Terence Tao: Using analytic number theory in unexpected ways

Regarding this method of "abandoning a child", Terence Tao also gave a more professional interpretation:

If we let (σ,) denote the number of zeros of the Riemann zeta function with a real part at least σ and an imaginary part at most , the Riemann hypothesis tells us that for any σ>1/2, (σ,) will vanish, although we cannot prove this unconditionally.

But as a next step, we can prove a non-trivial upper bound on the zero density estimate, i.e., (σ,).

It turns out that σ = 3/4 is a critical value. In 1940, Ingham came up with a result - (3/4,) ≪ ^{3/5 + (1)}.

Over the next eighty years, the only improvement to this bound was a small improvement in the error of (1).

This limits what we can do in analytic number theory: for example, in order to get a good prime number theorem on almost all short intervals of the form (,+^), we have long been restricted to >1/6, the main obstacle being the lack of improvements to the Ingham bound.

The latest research by Guth and Maynard successfully improved the Ingham limit from 3/5=0.6 to 13/25=0.52.

This led to many corresponding improvements in analytic number theory; for example, the range of prime number theorems that could be proved in almost all short intervals changed from >1/6=0.166… to >2/15=0.133… (if the Riemann hypothesis were true, this would mean that we could cover the entire range >0).

The arguments are essentially based on Fourier analysis. The first few steps are standard and will be recognizable to many analytic number theorists who have tried to break Ingham's bounds.

But they have many clever and unexpected manipulations, such as controlling the key phase matrix ^{}=^{log⁡} by raising it to the sixth power (which, on the surface, makes the problem more complicated and tricky).

and, refusing to use the stationary phase method to simplify a complicated Fourier integral, thereby yielding on the exponent in order to preserve a factorization form that ultimately proves more useful than the stationary phase approximation; and dividing the case according to whether the locations where the large values ​​of the Dirichlet series occur have small, medium, or large additive energies, and adopting a slightly different argument for each case.

Here, the exact form of the phase function log⁡ implicit in the Dirichlet series becomes very important; this is an unexpected way to exploit a special kind of exponential sums that arise in analytic number theory, rather than the more general exponential sums that one might encounter in harmonic analysis.

References:

https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/