"Using the regularity of the boundary points of the Dirichlet function to construct a function domain with regularity boundary, and then introducing the curve equation by expanding the field, the concept of dual reduction group is ......"

In the auditorium of Wenjin International Hotel, Artur Avila's eyes suddenly brightened after muttering a few words, and he looked at Xu Chuan excitedly.

"Xu, you really deserve to be known as the strongest genius in the history of mathematics, too powerful, using this method, maybe you can really restrain and determine the functivity of a part of the self-defending group."

Xu Chuan's face was embarrassed, what the hell is this 'strongest genius in the history of mathematics'? Who put this name on him?

However, during the exchange and discussion, he didn't pay much attention to this, nodded his head, and continued with Professor Artur Avila's words:

"Not only that, but the first instance of the Langlands functal conjecture to be verified is on the functality between the self-defending representation of GL2 on the algebraic number field and the representation of the multiplicative subgroup of quaternion algebra."

"The function demonstrated in this classic work also proposes the relationship between the original form of the Artin conjecture and the function conjecture, which is also reformulated as the functal conjecture between the two-dimensional complex representation of the Galois group and the GL2 self-respecting group representation."

"Thus, the Atin conjecture states that the Atin l functions constructed on the Gavaro group are all pure, while Langlands conjectures that these Atin l functions should essentially be L functions represented by the self-defended group."

Hearing this, Professor Artur Ávila fell into deep thought, but after a while, he suddenly came to his senses, and said with half confusion and half affirmation:

"If the Atin conjecture can be proved, then it will be a big step forward for the Atin l function on the Langlands conjecture?"

Xu Chuan nodded and said, "Judging from the current theory, this is indeed true. ”

Immediately, he shook his head again and said, "But ......."

"But it's too difficult to solve the Atin conjecture." Professor Artur Avila sighed and added what Xu Chuan had not finished.

Xu Chuan acquiesced and did not speak again.

The Artin conjecture, also known as the new Meissen conjecture, is a generalized derivation of the famous Masonne conjecture, which is a conjecture about prime numbers.

If you haven't heard of the Artin conjecture and the Mason conjecture, then most people who have heard of the familiar Goldbach conjecture should have heard of it.

They are all conjectures of the same type, and they can be said to be derived from prime numbers.

In mathematics, people first came into contact with natural numbers such as 0, 1, 2, 3, and 4.

And in such a natural number, if a number is greater than 1 and is not divisible by other natural numbers (other than 0), then this number is called a prime number, also called a prime number.

Numbers that are larger than 1, but not prime, are called composites, and 1 and 0 are special in that they are neither prime nor composite.

This peculiar phenomenon was noticed as early as 2,500 years ago, and Euclid, the father of geometry by the ancient Greek mathematician, proposed a very classic proof in his most famous work, Geometry Primitives.

That is, Euclid proved that there are infinitely many primes, and proposed that a small number of primes can be written in the form of "2^p-1", where the exponent p is also a prime.

This proof is known as Euclid's prime number theorem, and it is one of the most basic classical propositions in number theory.

Classics never go out of style, and subsequent mathematicians have derived a variety of conjectures about prime numbers when they studied the 'Euclidean prime number theorem'.

Starting from the Mersenne prime conjecture, to the Zhou guess, the twin prime conjecture, the Ulam spiral, and the Gilbreth conjecture........ to the eventually famous Goldbach conjecture and so on.

There are many conjectures derived from prime numbers, but most of them have not been proven.

The new Meisenne prime conjecture that Xu Chuan and Professor Artur Avila talked about is a conjecture derived from prime numbers, also known as the Artin conjecture, which is an upgraded version of the original Mersenne prime conjecture.

Among the many prime conjectures, the difficulty is comparable to the twin prime conjecture, second only to the famous 'Goldbach conjecture'.

[New Masonne Prime Conjecture: For any odd natural number p, if two of the following statements are true, the remaining sentences will be true:

1. p=(2^k)±1 or p=(4^k)±3

2. (2^p)-1 is a prime number (Mersenne prime number)

3. [(2^p)+1]/3 is a prime number (Wagstaff prime number)]

The New Merson prime conjecture has three problems, and the three problems are closely related, and if two of them can be proved, then the remaining one will naturally hold.

In the history of scientific development, the search for Mersenne primes has been used as an important indicator to measure the development of human intelligence in the era of manual arithmetic records.

Just like today's IQ test questions, the more Mersenne primes you can calculate, the smarter you are.

Although the Mersenne prime number seems simple, when the exponential p-value is large, its exploration requires not only advanced theory and skill, but also painstaking calculations.

The most famous, Euler, known as the "God of Mathematics", proved by mental arithmetic that 2^31-1 is the 8th Mersenne prime when he was blind;

This prime number with 10 digits (i.e., 2147483647) was the largest known prime number in the world at that time.

It is very good for ordinary people to be able to add, subtract, multiply and divide three-digit numbers, but Euler's mental arithmetic can push numbers to the billion-level level, and this terrifying calculation ability, brain reaction ability and problem-solving skills can be said to be worthy of the reputation of "the chosen child".

In addition, in 13 years, a research team led by Curtis Cooper, a mathematician at Central Missouri University in the United States, discovered the largest Mersenne prime to date - 2^57885161-1 (2 to the 57885161 power minus 1) by participating in a project called "Internet Mersenne Prime Search" (GIMPS).

This prime number is also the largest known prime number, with 17425170 digits, 4457081 digits more than the previously discovered Mersenne prime.

If it were printed in a normal eighteen-point standard font, it would be more than sixty-five kilometers long.

It's a big number, but in mathematics, it's very small.

Because 'number' is infinite, and number has the concept of infinity, mathematically speaking, after the number 2^57885161-1 (2 to the 57885161 power minus 1), no one knows how many prime numbers there are.

This thousand-year-old journey of exploration is the largest in the history of mathematics: how many Mersenne primes there are, and whether they are infinite, no one has been able to answer until now.

Proving the new Merson prime conjecture is no less difficult than the Weyl-Berry conjecture that Xu Chuan had proved before.

So far, the most difficult proof of prime conjecture in mathematics is only the weak Goldbach conjecture.

That is: [Any odd number greater than 7 can be expressed as the sum of three odd prime numbers.] 】

In May 2013, Harold Hoovgot, a researcher at the École Normale Supérieure in Paris, published two papers in which he announced that he had completely proved the weak Goldbach conjecture.

In addition, in the same year, Professor Zhang Yitang, a mathematician from Huaguo, also made considerable progress in the proof of the prime number conjecture.

His paper, "Bounded Distances Between Prime Numbers", was published in the Annals of Mathematics, which solved a puzzle that had plagued the mathematical community for a century and a half, and proved the weakening of the twin element conjecture.

That is, it is found that there are infinitely many pairs of prime numbers with a difference of less than 70 million.

This is the first time that it has been shown that there are an infinite number of pairs of prime numbers with a spacing less than a fixed value.

But for the mathematical community, both the weak Goldbach conjecture and the weak twin prime theorem are just a prelude to climbing the peak.

They are like a resounding national anthem of a climber who climbs Mount Everest before setting off, which can give climbers courage to a certain extent, but it is unrealistic to expect to climb Mount Everest and stand on the summit.

........

"Xu, will you try to develop in the direction of number theory?"

After a slight silence in the atmosphere, Professor Artur Ávila looked up at Xu Chuan.

If this youngest genius in the history of mathematics develops in the direction of number theory, maybe he will have a chance to pick a huge fruit in the field of prime numbers?

He didn't dare to say for sure, after all, who could be sure of such a thing.

Artur Avila would love to see the day Goldbach's conjecture was confirmed, but he didn't want the rising star of mathematics in front of him to dive headlong for years or even decades without results.

Prime numbers have been developed for thousands of years, and countless mathematicians have rushed into this huge pit one after another, although they have proved many conjectures and solved many problems.

But all the while, the hardest problems were never solved.

Even, there is no hope of a solution.

However, if Xu Chuan continues to study spectral theory, functional analysis, and Dirichlet functions, he does not dare to say that he will definitely be able to make greater contributions than the Weyl-Berry conjecture, but he will definitely be able to further expand the boundaries and expand the scope of mathematics in these fields.

If you can transfer to number theory, you are not sure.

Not every genius is Tao Zhexuan's, at present, Xu Chuan's mathematical talent is indeed higher than Tao Zhexuan's, but no one knows what will happen after crossing fields.

..........

Xu Chuan did not give Avila a precise answer, in the past year, he did read a lot of books related to number theory, but number theory is not in his follow-up study and research arrangement.

He prefers to be able to solve physical problems with practical functions and analysis, while number theory mainly studies the properties of integers, which can be regarded as pure mathematics.

Of course, with the development of mathematics to this day, it cannot be said that any field of mathematics is pure mathematics, and it can always be linked to other fields.

For example, in statistical mechanics, the distribution and fraction function is the basic mathematical object of research; In the analytic theory of prime number distributions, the zeta function is the basic object.

Thus, this unorthodox interpretation of the zeta function as a partition function points to a possible fundamental connection between the distribution of prime numbers and this branch of physics.

However, at present, there is still a gap in the application of number theory to the field of physics, which is far less extensive than the fields of mathematical analysis, function transformation, and mathematical modeling.

Therefore, Xu Chuan is not very inclined to devote a lot of energy and time to the field of pure number theory.

But it's a matter of studying number theory.

Because number theory is not only pure number theory, but also various branches such as analytic number theory, algebraic number theory, geometric number theory, computational number theory, arithmetic algebraic geometry, etc.

These branches are all extensions of pure number theory, that is, elementary number theory, combined with other mathematics.

For example, analytic number theory is the study of number theory about integer problems with the help of calculus and complex analysis (i.e., complex variable functions).

The things he talked about with Professor Avila tonight have something to do with analytic number theory.

Because in addition to the circle method, sieve method, etc., the analytic number theory methods also include the modular form theory related to elliptic curves, and so on. Later, it developed into the theory of self-contained forms, which was linked to the theory of representation.

Therefore, having a certain foundation in number theory is still of great help to other mathematics learning.

.......。