The $1 Million Mathematical Riddle

Reimann Zeta Function
Reimann Zeta Function

There are some constant numbers in physics and mathematics. Even these fixed numbers are called transcendental or transcendental numbers. Numbers such as π, e, and φ often appear in unexpected places in science and mathematics. Pascal's triangle and the Fibonacci sequence are also inexplicably common in nature. In addition to these, we can count the Riemann zeta function, a deceptively simple function that has confused mathematicians since the 19th century. The most famous dilemma, the Riemann hypothesis, is perhaps the biggest unsolved question in mathematics, with the Clay Mathematics Institute offering a $1 million prize for a correct proof. As we stated in the title of our article, that is, “The 1 Million Dollars Mathematical Riddle”.

UC Santa Barbara physicist Grant Remmen believes he has a new approach to exploring the quirks of the zeta function, and he's done some work on the topic as well. He found an analogue that translated many of the important properties of the function into quantum field theory.

This means researchers can now take advantage of tools in this field of physics to probe the enigmatic and strangely ubiquitous zeta function. His work holds the possibility that it may even lead to a proof of the Riemann hypothesis.

Remmen lays out his approach in the journal Physical Review Letters. There is an interesting similarity between his work and his name. "The Riemann zeta function is the famous and mysterious mathematical function that pops up everywhere in number theory," said researcher Remmen. “It has been researched for over 150 years,” he explains.

As a principal physics officer at UC Santa Barbara, Remmen normally devotes his attention to topics such as particle physics, quantum gravity, string theory, and black holes. "In modern high energy theory, both the physics of the largest scales and the physics of the smallest scales hold the deepest mysteries," he said.

Most people understand quantum mechanics (subatomic particles, uncertainty, etc.) and special relativity (time dilation, E = mc).2 etc.) heard.

“But with quantum field theory, physicists understood special relativity and quantum mechanics, how particles moving at or near the speed of light behave and how to relate things to one another,” he explained.

Quantum field theory is not exactly a single theory. It's more like a collection of tools that scientists can use to identify any set of particle interactions.

Remmen realized that one of the concepts here shares many properties with the Riemann zeta function. This is called the scattering amplitude.

It encodes the quantum mechanical probability of particles interacting with each other. This subject intrigued him.

Scattering amplitudes are usually expressed as complex numbers. These numbers consist of a real part and an imaginary part – a multiple of √-1, which mathematicians call i. Scattering amplitudes have nice properties in the complex plane. First, they are analytic (which can be expressed as a series) around every point except for a particular set of poles, all of which lie along a line.

Reimann Zeta
Reimann Zeta Function Frequency Values

“This thought there was a similarity going over the zeros of the Riemann zeta function,” Remmen said.

I thought about how I could determine if this apparent resemblance was real.”

The scattering amplitude poles correspond to particle generation in which a physical event occurs that creates a particle with momentum.

The value of each pole corresponds to the mass of the created particle. So it was a matter of finding a function that acts as the scattering amplitude and whose poles correspond to the trivial zeros of the zeta function. With pen, paper, and a computer to check his results, Remmen set about designing a function with all the relevant features. “I had the idea of ​​linking the Riemann zeta function to amplitudes for several years,” he said. “Once I started finding such a function, it took me about a week to build it and several months to fully explore its features and write the article.”

It Could Be a Surprisingly Simple Reduced Solution

In essence, the zeta function generalizes the harmonic series.

This series explodes to infinity when x ≤ 1, but converges to a real number for every x > 1.

In 1859, Bernhard Riemann decided to consider what happens when x is a complex number.

Riemann also decided to extend the zeta function to numbers where the real component is not greater than 1 by defining it in two parts.

Thanks to a theorem in complex analysis, mathematicians know that there is only one formulation for this new field that preserves the properties of the original function seamlessly. Unfortunately, no one has been able to represent it in a finite number of terms that are part of the mystery surrounding this function.

Given the simplicity of the function, it should have some nice features. When the zeta function is written in certain formats, all negative even numbers map to zero, whether this is obvious or "trivial" as mathematicians call it. What confuses mathematicians is that all other trivial zeros appear to lie along a line: each has an actual ½ component.

Riemann assumed that this pattern holds for all these trivial zeros, and the trend is confirmed for the first few trillions of them. However, there are assumptions that work for trillions of examples and then fail in very large numbers. That is, mathematicians cannot be sure that the hypothesis is true until it is proven.

But if true, the Riemann hypothesis has far-reaching implications. “It pops up everywhere in fundamental questions in math for a variety of reasons,” Remmen said. Assumptions in different fields, such as theory of computation, abstract algebra, and number theory, depend on keeping the hypothesis true. For example, proving this will provide an accurate account of the distribution of prime numbers.

A Physical Analog

The scattering amplitude that Remmen found describes two massless particles interacting by changing an infinite set of massive particles, one at a time. There is a point where the function cannot be expressed as a series - a pole - corresponding to the mass of each intermediate particle. Together, the infinite poles align with the non-trivial zeros of the Riemann zeta function.

What Remmen is building is the leading component of interaction. There are an infinite number of processes, each of which takes into account smaller and smaller aspects of the interaction and describes processes involving the exchange of more than one large particle at the same time. These “cycle-level amplitudes” will be the subject of future studies.

The Riemann hypothesis assumes that all non-trivial zeros of the zeta function have a real component of ½. Let's translate this into Remmen's model: All poles of the amplitude are real numbers. This means that if someone can prove that the function describes a coherent quantum field theory—that is, a theory in which masses are real numbers, not imaginary—then the Riemann hypothesis will be proven.

This formulation brings the Riemann hypothesis to another field of science and mathematics that has powerful tools to present to mathematicians. “Not only is this association with the Riemann hypothesis, but there is a list of other properties of the Riemann zeta function that correspond to something physical in the scattering amplitude,” Remmen said. For example, he has already discovered non-intuitive mathematical identities related to the zeta function using the methods of physics.

Remmen's work follows the tradition of researchers looking to physics to shed light on mathematical dilemmas. For example, physicist Gabriele Veneziano asked a similar question in 1968: whether the Euler beta function can be interpreted as a scattering amplitude. “It might indeed be,” Remmen said, “and the amplitude that Veneziano created was one of the first string theory amplitudes.”

Remmen hopes to take advantage of this amplitude to learn more about zeta function. “Having all these analogs means there is something going on here,” he said.

If we collect our word, we will not give up on bringing a new change to you every day.

source: physorg

Günceleme: 07/03/2022 21:40

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