249: Chapter 249 Goldbach - Zhao Theorem

The next day.

This was the final day of this session of the International Congress of Mathematicians after it had been forcibly postponed.

Before the start of the final keynote presentation.

Inside the largest Lecture Hall No. 1 of the International Convention Center.

At this time, the audience was already packed. The venue had a rated capacity of four thousand seats, but today at least five thousand people had flooded in.

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Almost all the mathematicians who came to attend this congress were gathered in this hall. Even the surrounding aisles, the steps of the main passage, and even the base of the back walls were densely packed with people standing.

The air was filled with a sense of stuffiness caused by the extreme crowding.

Everyone was staring in the direction of the rostrum, talking in low voices. The venue was filled with the buzzing sound of various languages mixed together; it was the mathematicians sharing their insights from reading the paper yesterday.

Nine o'clock in the morning sharp.

The exclusive side door on one side of the lecture hall was pushed open.

Zhao Yang walked into the venue with a calm expression.

Following the red-carpeted aisle, under the gaze of thousands of people, he walked directly onto the podium.

The moment Zhao Yang appeared, the originally noisy lecture hall instantly fell silent.

Thousands of pairs of eyes looked in unison at Zhao Yang on the stage.

They were the smartest handful of people on this planet, and at this moment, their eyes were filled with anticipation as they looked at Zhao Yang.

Zhao Yang walked to the microphone on the podium.

He did not give any self-introduction.

"I presume you have already read the preprint of my paper yesterday, and many of you have questions regarding it."

Zhao Yang spoke to all the mathematicians below in a very calm tone.

"I will first go over the core of my paper, and then I will give everyone time to ask questions."

After saying this, Zhao Yang turned around and picked up a piece of chalk.

He faced the giant blackboard.

Turning his head toward the microphone on the podium, he said calmly:

"Keep up with my train of thought!"

Having said that, Zhao Yang began writing on the blackboard while simultaneously explaining.

"Let N be a sufficiently large even number, and we define the objective function as R(N)=∑p1+p2=N log p1 log p2..."

Without any preamble, Zhao Yang dove straight into the most core area of the proof.

The chalk in his hand moved rapidly across the blackboard as the core parts of the proof for Goldbach's Conjecture were continuously outputted.

All the mathematicians below watched this scene with serious expressions.

In the VIP seats of the first row.

Professor Deligne frowned, staring at Zhao Yang's derivation on the blackboard.

He turned his head and asked in a lowered voice to Professor Faltings sitting next to him, "Gerd, you've read the paper. Do you think Zhao Yang has solved this problem?"

Professor Faltings was famously rigorous and stiff; he did not answer immediately.

He pushed up the glasses on the bridge of his nose and remained silent for a moment.

"Highly likely. At least I haven't found any fatal flaws in the proof."

Professor Faltings gave an extremely conservative evaluation.

"His entire Topological Manifold framework is very solid, with no logical flaws. However, I still have some questions about his final step using Automorphic Forms to handle the distribution of zeros. He took too large a leap, lacking some necessary lemma support."

Professor Deligne nodded and continued to focus his attention on the stage.

Time passed minute by minute.

Gradually, as the formulas Zhao Yang wrote on the blackboard became deeper and the dimensions involved became higher...

A clear stratification began to appear in the venue.

Those scholars engaged in applied mathematics, partial differential equations, or general combinatorics began to fall behind in large numbers. Looking at those Algebraic Geometry symbols on the blackboard, their brains were completely unable to keep up.

But the top Number Theory experts and Algebraic Geometry masters sitting in the front row could still barely keep up with Zhao Yang's terrifyingly fast speed of deduction. Their breathing became somewhat rapid, and their eyes grew brighter and brighter.

Fifty-five minutes later.

Zhao Yang finished writing the last upper bound estimation inequality for integral convergence.

"...Therefore, the error term tends to zero, and the main term is greater than zero. The conclusion holds."

Zhao Yang accurately tossed the half-piece of chalk in his hand into the chalk box and brushed the dust off his hands.

He turned around to face the dense crowd below.

"My proof ends here. Does anyone have any questions?"

Zhao Yang asked calmly.

The entire hall was silent for about ten seconds.

Soon, Professor Faltings, sitting in the first row, raised his hand.

The staff immediately handed him a microphone.

Faltings stood up, looking at Zhao Yang with extremely sharp eyes.

"Zhao, regarding your derivation in the third section on the right side of the blackboard. You utilized the Langlands Program for local-to-global mapping. But when dealing with integrals on the Adele Group, how can you guarantee that the product of Local factors at Non-Archimedean places is absolutely convergent? You didn't provide a strict definition in the preprint."

This was a question that hit the nail on the head.

The eyes of the entire hall focused on Zhao Yang once again.

Zhao Yang didn't panic at all; after a moment of thought, he quickly provided an answer.

"This is a very basic estimation omission; I omitted it in the preprint to save space."

Zhao Yang picked up the chalk, turned around, and quickly wrote down three extremely brief lines of lemma derivation in a corner of the blackboard.

"Professor Faltings, according to the known results of the Ramanujan-Petersson Conjecture, for the Fourier coefficients of Cusp forms, we have |ap| ≤ 2p^((k-1)/2). By substituting this upper bound into the Euler product, you can directly use the domain of absolute convergence of the Dirichlet series to reach the conclusion. No additional definition is required."

Faltings stared at those three lines on the blackboard, his brain working at high speed.

A few seconds later, a hint of satisfaction appeared on his extremely serious face.

He nodded and sat back down without asking further questions.

Seeing Faltings convinced, a low commotion broke out in the venue.

Immediately following.

Shing-Tung Yau also picked up the microphone and began to ask questions about the metrics of Calabi-Yau manifolds in high-dimensional topological spaces.

Zhao Yang likewise didn't pause for a moment, accurately pointing out the isomorphic relationship between geometric boundaries and algebraic mappings, easily resolving Shing-Tung Yau's doubts.

In the following half hour.

Several top professors from major universities took turns, throwing out all sorts of extremely tricky questions.

Standing on the stage, facing these top mathematicians' frantic search for loopholes, Zhao Yang responded with a calm expression. With his current IQ, dealing with these questions could be said to be more than easy!

Finally.

When the last professor who asked a question sat down, the entire venue fell into a dead silence.

No one raised their hand anymore; Zhao Yang had already convinced these smartest brains in the world. These top mathematicians had basically all understood Zhao Yang's proof logic.

All within just that past hour!

Professor Deligne, sitting in the center of the first row, stood up.

He was nearly seventy years old this year. He looked at that excessively young Chinese scholar on the stage, his eyes filled with endless admiration.

Professor Deligne took the microphone, his voice echoing in the massive lecture hall.

"After rigorous academic review and on-site defense, I now, on behalf of myself and my colleagues present, can provide an objective conclusion."

Professor Deligne paused, his gaze sweeping across the entire hall.

"Zhao Yang's proof is logically rigorous with no fatal flaws. He has solved Goldbach's Conjecture—oh no, I think it should now be called the Goldbach-Zhao Theorem!"