The top student must be diligent

Xiao Yi began to observe this equation.

This is the most important result of Artin's conjecture.

It is this functional equation that makes Artin's conjecture true: every finite-dimensional complex representation ρ:Gal(K/k)→GL(n,C) should correspond to an automorphic representation π, so that their L-functions are equal: L(s,ρ)=L(s,π).

Through this result, it is even possible to study the functor conjecture.

Of course, Xiao Yi's research focus is not the functor conjecture now.

Now, he wants to see how to connect this formula with the Riemann hypothesis.

Soon, he smiled slightly, and the pen in his hand moved again.

Since it has come to this point, Artin's conjecture has been proved by him, and the difficulty of the next step can no longer be difficult for him.

Although he still has to deal with a series of quite complicated derivations, which may take a long time, it is certain that it is no longer difficult for him.

…

Time flies again.

About a month has passed.

During this month, the world is still the same as it should be, and no changes have taken place.

Of course, for China, the most important thing is that several nuclear fusion power stations have been built and put into operation.

As the initial time passes, the next period is indeed the time when nuclear fusion power stations are produced in large quantities.

Those core economic zones have basically used electricity from nuclear fusion, which has brought a grand bull market to the A-share market. This bull market has almost never stopped since the beginning of this year.

At the beginning, stock investors were still a little bit unconvinced. After all, recalling the last bull market, it was in 2024. The bull market was booming for a few days, and then it caused a heavy blow to the crazy stock investors.

However, when this bull market rose for a week in a row, they began to doubt it, and after rising for half a month in a row, they had to try to invest in it.

Until a month of continuous growth, finally, stock investors fell into madness again.

Until now, the stock market is still rising, and it has hardly stopped.

The market has reached an unprecedented 9,500, which has long exceeded the 6,124 points set in 2007 by more than half, and it is just around the corner to break through 10,000.

This is mainly because nuclear fusion energy has reduced the operating costs of many companies, especially for those industrial companies. It should be noted that China has the most industrial companies, and China has always attached importance to the development of the industrial economy. It has never vigorously developed the financial economy like those Western countries, while ignoring the development of the industrial economy, especially in industry.

China's industry can be said to be the biggest beneficiary after the realization of nuclear fusion.

As for the mathematicians of the Prime Number Pioneer Project, they are still troubled by what direction they should choose. Regarding this issue, they even had a lot of disagreements at the beginning, and these disagreements also made them decide to split into several teams, each conducting research from different directions, and then regularly exchanging results, and then judging which direction has more opportunities based on these results.

Based on this approach, they have achieved some results.

For example, those mathematicians who want to continue to develop the critical line theorem continue to study based on the critical line theorem, and now they have successfully come up with a higher number than the 62.5% given by Xiao Yi at the beginning, 66.67%, which is almost two-thirds.

However, by the same token, two-thirds seems to be very close to the final answer, but in fact it is not, and there is still a distance like a natural moat.

Even their current results can only be said to be developed on the basis of Xiao Yi's original results, and cannot be said to have achieved results that are worth celebrating.

As for mathematicians in other directions, they have also made certain breakthroughs to a greater or lesser extent, but these breakthroughs cannot be called much. If they want to publish papers, I am afraid they may not be enough for the first district - perhaps relying on the reputation of their group of bigwigs, the editors may agree to publish their papers in the first district for their sake.

However, these great mathematicians basically couldn't afford to lose face, so they finally created a website called the Prime Number Pioneer Project, and then published all their achievements directly on this website so that people can see where they have gone.

Of course, it is precisely because of this that the media have been saying all day that Xiao Yi did not publish his research progress like them, and used this to mock Xiao Yi for not making any progress, or that he did not publish his research results because he was worried about failure, and then kept repeating what others said.

Although Xiao Yi basically did not see these sarcasms, even if he did, he did not care.

In this way, time came to the early morning of July 15th.

…

[For any CM elliptic curve E, there exists a generalized modular curve X and an embedding i: E→ X, such that i induces an isomorphism between Hecke characteristics: λ_Eλ_X i_*, where λ_X is the Hecke characteristic of X and i_* is a homomorphism between Galois groups induced by i. ]

[Therefore, substituting into Theorem 8.9 and Theorem 9.1, we can determine that all zeros of L(s, E) are on the line Re(s)=1/2. ]

[Therefore, all non-trivial zeros of ζ(s) are also on the line Re(s)=1/2. ]

[In summary, the Riemann hypothesis holds. ]

The final proof is complete.

The pen in Xiao Yi's hand also stopped at the last period at this moment, and did not leave for a long time, as if condensing the passage of time.

Riemann hypothesis.

Riemann hypothesis.

Riemann hypothesis.

Riemann theorem!

…

Any famous mathematical conjecture has a different history.

But no conjecture has such an extraordinary status as the Riemann hypothesis.

At this moment, history and the present intersect, and the problem that countless mathematicians have struggled for, paid for, and devoted their entire lives to in the past has come to an end in his pen.

Countless pictures seemed to pass through his mind.

In his office, Bernhard Riemann wrote the letter entitled "On the Number of Primes Less Than a Given Value" to express his return for the high honor of becoming a member of the Berlin Academy of Sciences. At that time, he probably didn't expect that his paper of only eight pages would become the starting point of the Riemann hypothesis that almost all mathematicians are obsessed with.

He seemed to see that generations of mathematicians have been thinking, arguing, and exploring this problem one after another.

Whether it was Euclid thousands of years ago, or Euler, Gauss, Hardy later...

Until now, mathematicians such as Selberg, Bombieri, Faltings, Deligne, etc.

These names have become the guiding lights to the answer to this question, until now.

The young man named Xiao Yi finally lit up the last lamp to the truth.

The pen in his hand finally stopped standing and was gently put aside by him.

He got up, stretched his waist, walked to the window, and opened the curtains.

The light of the early morning shone in.

Last night, he could be said to have not slept all night.

But at least, this night was not in vain.

"However, the buff level has not been upgraded..."

In response to this, Xiao Yi could only shake his head helplessly. At this point, the buff level was not so easy to upgrade.

As for whether it might be because his proof was wrong, that was completely impossible. He had sufficient confidence in his proof.

So, he probably needs to solve another problem of similar level before the buff can be upgraded?

This question quickly passed through his mind, and now he didn't want to think about it anymore.

After stretching for a while, he yawned, then returned to his seat and reviewed the previous proofs.

It was just a routine inspection, but this time, he found an unexpected idea from these proofs.

"Automorphic representation... L-function... and some kind of 'natural' object in geometry?"

"Is it... for each automorphic representation ρ, we can construct a number theory L-function L(s, ρ), and a geometric "natural" object X(ρ)..."

He took out the pen again, and then wrote an equation on it, muttering: "So that they all satisfy such a relationship."

[L(s, ρ) = L(s, X(ρ))]

That is, the L-function of ρ is equal to some "natural" L-function of X(ρ).

He put down the pen again and hugged his head.

If this is true, then it will be amazing.

This means that he has achieved a great generalization based on the Langlands Program.

The Langlands Program foresees that each automorphic representation should correspond to a geometric object and an L-function in number theory.

And this relationship further foresees that there should be a direct equation relationship between this geometric object and the L-function.

And such a relationship is extremely important for mathematics.

It provides a new unified perspective, linking the three major branches of mathematics, algebra, geometry, and analysis, so that mathematicians can transform problems in algebra into problems in geometry or analysis!

But is this equation really possible?

Xiao Yi doesn't know.

Because this is a brand new problem.

It takes a long proof process.

But now Xiao Yi doesn't want to think too much anymore.

In the next week, he just wants to give himself a vacation.

The Riemann hypothesis has been proved for so long, can't you enjoy it?

Of course you can.

"As for this new problem, then..."

"Let's name it the Xiao conjecture."

Well, he proved the Artin conjecture and the Riemann conjecture, and now he will give the mathematics community an even more powerful conjecture.