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Chapter 150: Chapter 148: Professor Bombelli, Please Ask

When Enrico Bombelli heard the paper was authored by Li Dong, he fell silent.

An hour ago, it would have been a different story.

Even if this somewhat unorthodox paper had been signed by Li Dong, he wouldn’t have given it a second glance and would have just told the editor to reject it outright.

Montgomery’s Conjecture on Association! It was the divine bridge that connected the distribution of the Riemann ζ-function’s zeros to the Gaussian Unitary Ensemble (GUE) of random matrix theory! It was the core link between prime number distribution and quantum chaos!

Since Montgomery proved the case for |α|

And now you were telling Bombelli that you’d pushed it to |α| ∈ [0, 4]? That was pure fantasy!

But...

Just moments ago, he had personally witnessed Li Dong’s 45-minute invited academic report on stage.

That young man, using the Riemann explicit formula as his core, combined it with fine-grained estimations from the prime number theorem and a Fourier optimization framework. He had managed to perfectly contain the remainder terms contributed by high-order powers of primes within the convergence bounds!

The logical deduction was so beautiful, and incredibly rigorous.

It was as if God himself had guided his hand as he wrote on the blackboard, without a single flaw.

How could someone so rigorous in his academic work possibly submit a trash paper that was pure nonsense?

After a brief silence, Bombelli spoke slowly.

”Send it to my email. I’ll take a look.”

The academic editor on the other end of the line quickly replied.

”Professor, it’s already in your inbox.”

Bombelli hung up the phone.

He took a tablet from the briefcase beside him and opened his work email.

The first email in his inbox contained the PDF attachment.

”A Proof of Montgomery’s Conjecture on Association in the Interval |α| ∈ [0, 4]”

Bombelli opened the paper.

At that moment, the entire lecture hall was still in an uproar over Li Dong’s refusal to release his algorithm.

Wang Zhiqiang, leading a group from Capital University and a band of Jiang Yubai’s followers, was righteously engaged in moral blackmail, while a group of professors led by Liu Ruochuan was arguing back just as heatedly.

But all this noise had completely faded away in Bombelli’s world.

”...Based on a massive sample dataset of the first 10^23 non-trivial zeros of the Riemann zeta function, and assuming the Riemann Hypothesis holds true... this paper extends the applicable range of Montgomery’s theorem from the original |α|

Bombelli read on, line by line.

As the derivations deepened, the eighty-year-old Fields Medal laureate’s brow furrowed tighter and tighter.

He found that he... couldn’t quite follow it!

It wasn’t that Li Dong’s writing was chaotic, but that the logical leaps were simply too vast!

When advancing the interval to |α| ∈ [2, 3] and [3, 4] and even higher, Li Dong had separated the main term of the Riemann explicit formula.

For the interval |α| ∈ [2, 3], he controlled the upper bound of the remainder term using only the prime number theorem. Then, for the boundary interval of |α| ∈ [3, 4], he went even further, introducing a Fourier optimization framework to completely suppress the risk of divergence from the remainder terms of high-order powers of primes!

’This error scaling... how did he ensure it doesn’t diverge along the integration path of this high-order prime remainder term?’

’And the construction of this Fourier optimization weight function—when deriving to the boundary |α| = 4, why can it guarantee the remainder term’s decay rate perfectly matches the main term’s convergence requirement?’

Bombelli continued reading.

Although there were some details in the scaling that he couldn’t understand.

After all, with mathematical papers at this level, it wasn’t always possible to see through every step at a glance.

It required the author to explain it personally, to be questioned during peer review and in subsequent seminars before it could be fully understood.

However, relying on the intuition he had honed over a lifetime of studying analytic number theory, he followed Li Dong’s overarching logic and mathematical framework and deduced...

He couldn’t believe it himself. The paper’s derivation, on a macro-level, was completely self-consistent!

Not a single contradiction! Not a single paradox!

The more Bombelli read, the quieter he became, not saying a word.

The paper was simple in a sense. It didn’t prove Montgomery’s conjecture; it only pushed the boundary, which had been stuck at |α|

But that alone was terrifying enough! Because once the boundary was pushed that far, the number of things it could be used to prove was simply staggering...

Bombelli stared, without blinking, at a sentence in the paper’s introduction.

”...For the first time, it reveals the deep connection between the pair correlation statistical properties of automorphic L-function zeros and the local ramification index.”

’My God...’ Bombelli exclaimed in his mind.

’If the boundary is pushed to 4, it means this isn’t just a festival for analytic number theory!’

’This set of theoretical tools for pair correlation will directly provide a brand-new, computable numerical criterion for the local-global compatibility of automorphic representations!’

He thought of his colleagues in the field of pure algebra, who had grown gray-haired struggling with the Langlands Program.

Previously, they could only handle low-ramification cases of GL(2) using complex pure algebra methods, and were helpless in high-dimensional, high-ramification scenarios.

But now, Li Dong had pushed the boundary to 4. As long as this paper held up, the Langlands correspondence would gain its first-ever analytically verifiable tool, directly breaking through the technical bottleneck of traditional algebraic methods in high-dimensional GL(n) cases!

It was equivalent to taking the academic community’s achievements from a single-story house and building a skyscraper!

Gradually, the eighty-year-old man’s chest began to heave violently.

”Professor! What’s wrong?”

His assistant, who had been sitting next to him and watching him closely, saw this and was scared out of his wits.

The old professor had a heart condition. The assistant immediately pulled a pill from his suit pocket and turned to ask a staff member for a cup of warm water.

”Professor, quick! Take your medicine!”

After swallowing the pill with the warm water and leaning back in his chair for several minutes, Bombelli gradually calmed down.

He looked up at Li Dong on the stage, his expression impossibly complex.

As an old-school scholar, Bombelli inwardly agreed with Li Dong’s decision not to disclose the underlying algorithm.

In this academic circle, who didn’t have a few undisclosed things up their sleeve?

Who in this circle didn’t see through the trick of ”openly robbing” someone under the grand banner of ”sharing knowledge for all humanity”?

But now, after reading this paper, Bombelli’s conviction wavered.

The paper’s macro-level logic was perfect. As long as it passed cross-validation and defense by top experts, it was very likely to be proven true.

But that presented a problem.

In the paper, Li Dong cited a massive sample dataset of 10^23 non-trivial zeros!

But the new dimensional reduction algorithm used to calculate this dataset hadn’t been disclosed by Li Dong at all. To the outside world, it was a complete black box!

According to the ironclad rules of top international journals, no matter how stunning the paper was, it could never be published in the *Annals of Mathematics* because its data could not be replicated by the academic community.

If the black-box data prevented it from being published in the *Annals of Mathematics*, would Li Dong simply choose not to publish it at all?

For the entire field of mathematics, that would be an immense loss.

The entire study of the Langlands Program would lose a near-perfect tool for advancement!

Even if he, Enrico Bombelli, were to cast aside his old face and ”steal” Li Dong’s paper to force progress for the sake of mathematics, it would be impossible.

Because he couldn’t understand the details either!

Without the author’s personal explanation, the highly non-linear Fourier optimization derivation in the paper was simply incomprehensible. Only the author himself knew how those extremely high-order prime remainder terms were so completely suppressed within the bounds.

Bombelli slowly closed his tablet.

He sighed and, in the end, slowly raised his hand.

The once-raucous hall fell silent upon seeing him raise his hand.

After all, the seniority and status of this living fossil of analytic number theory were undeniable.

Li Dong, standing at the podium, also glanced at the elderly man with some curiosity, then asked.

”Professor Bombelli, do you have a question?”

Under everyone’s gaze, Bombelli stood up, trembling slightly.

”Professor, you can ask from your seat!”

Seeing this, Li Dong quickly tried to dissuade him.

However, Bombelli shook his head, insisting on standing straight to ask his question.

This was his reverence for knowledge, and also his highest respect for the young scholar before him who could write such a revolutionary paper.

”Mr. Li.”

”A question regarding Montgomery’s Conjecture on Association.”

He didn’t mention that Li Dong had submitted it to the *Annals of Mathematics*, as that would violate his professional ethics as a special editor.

”May I ask for your guidance on it?”

As Professor Bombelli’s words fell,

The entire lecture hall was silent for a few seconds.

Then—

BOOM!

The whole venue was completely stunned.

”Montgomery’s Conjecture on Association?”

”Why would Professor Bombelli suddenly bring that up?”

Standing on the stage, Li Dong was also slightly taken aback. He hadn’t expected the editor from the *Annals of Mathematics* to forward the paper to Professor Bombelli so quickly.

After a moment of surprise, a confident smile appeared on Li Dong’s face.

”Professor Bombelli, please ask.”