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Chapter 157: Chapter 154: Your Approach Is a Bit Like the 19th-Century Masters

The approach was so classical it reminded Tao Zhexuan of the mathematicians from the era of Riemann and Cauchy.

Back then, there were no supercomputers, no Mathematica, not even electronic calculators.

Those men could only rely on pure mathematical intuition and geometric imagination to find the one true path through the infinite-dimensional maze of the complex plane, all by hand.

And Li Dong, it seemed, was one of them.

Suppressing his shock, Tao Zhexuan moved on to his next question.

”Then, in step four... your Fourier optimization framework for the boundary interval |α|∈[3, 4]—the way you constructed the weight function φ(x) was quite ingenious.”

”But I noticed that when you used this weight function to suppress the divergence of the remainder term from the p^3-order prime powers, there’s a leap in your chain of estimates.”

”To be specific, in going from the L²-norm estimate of the remainder to the final pointwise uniform convergence, you directly cited what appears to be a custom Sobolev embedding inequality, but the paper doesn’t provide a complete proof.”

”This inequality... did you derive it yourself?”

Li Dong put down his pen and thought for a moment before speaking.

”Yes, that inequality isn’t a standard result from the literature.”

”The proof is actually quite short; it can be done in about three or four steps.”

”The core idea is to use the weak-type estimate of the Hardy-Littlewood maximal function, coupled with a simple scaling argument on the measure of the weight function’s support set.”

”Simply put, you first use the maximal function to control the amplitude of the local oscillations, and then leverage the compact support of φ(x) in the frequency domain to upgrade the weak-type estimate to a strong-type one.”

Li Dong quickly jotted down a few lines of the derivation on a piece of notepad paper.

The process was clean and concise, the logic airtight.

After looking it over, Tao Zhexuan fell silent for a long moment.

The derivation itself wasn’t complex.

The issue, however, was that the path he had chosen was just so unconventional.

When faced with the leap from L² to pointwise convergence, a typical mathematician’s first instinct would be to look up the established frameworks of the classical Sobolev embedding theorem or Carleson’s theorem, then try to apply them.

But Li Dong’s approach was completely different.

He had bypassed all the ready-made theorems, starting from the most fundamental level with the Hardy-Littlewood maximal function, and forged an entirely new path using the most basic analytical tools.

It was like a modern person wanting to cross a river—their first instinct is to find a bridge or build a boat.

But not Li Dong. He just jumped straight into the water—and swam faster than a boat.

”This approach of yours...”

Tao Zhexuan chose his words carefully.

”If I may say something a bit unorthodox, your thought process when handling these refined estimates isn’t much like that of a modern mathematician.”

”It’s more like... one of the great masters from the nineteenth century.”

Tao Zhexuan looked at Li Dong.

”Your entire paper has that same quality.”

Hearing this, Li Dong’s heart skipped a beat.

’Seriously, how are you this perceptive?’

He certainly couldn’t tell Professor Tao that he had a WeChat group on his phone with *actual* nineteenth-century masters in it.

”Haha...” Li Dong gave a dry laugh, not daring to explain any further.

Tao Zhexuan nodded but didn’t press the issue.

He then asked several more technical questions about the Remainder Convergence Order, and Li Dong answered each one.

The logic behind every answer was airtight and irrefutable.

About twenty minutes later, Tao Zhexuan placed the manuscript back in his portfolio.

He leaned back against the sofa, his voice tinged with an almost helpless sense of admiration.

”To be honest, with this paper of yours... I can’t find anything that needs revision.”

Generally, even with a seemingly perfect paper, a reviewer can always find some phrasing to polish, a citation to add, or some formatting detail to optimize.

So, to find absolutely nothing to change... was extremely rare in the history of peer review for top-tier journals.

Tao Zhexuan paused, then spoke again as if remembering something.

”By the way, there’s one more thing.”

”Professor Nicholas Katz knows I’m in Shanghai, and he also knew I’d be meeting with you.”

”He called me and asked me to pass a message on to you.”

”By the way, Professor Katz is a key figure on the editorial board of the *Mathematical Annals*,” Tao Zhexuan said, his expression a little odd.

”Professor Katz said that the requirement Professor Bombelli raised at the conference regarding data reproducibility was based on a bit of a misjudgment.”

”Professor Bombelli was probably reading it rather quickly and misunderstood the nature of that set of data on the 10^23-order zeros, thinking it was a core part of the proof. That’s why he requested that you publish the algorithm.”

”But after thoroughly reading the paper, Professor Katz has confirmed that the data was merely for auxiliary verification, not an irreplaceable link in the chain of the proof.”

At this, Tao Zhexuan couldn’t help but chuckle.

”Professor Katz is... how should I put it? He feels the whole situation is very regrettable.”

”Because what Professor Bombelli said at the conference may have given you a poor impression of the *Mathematical Annals*.”

”So he hopes you’ll reconsider and let *Annals* publish the paper.”

The reason Tao Zhexuan chuckled was because...

Katz’s actual words had been far more scathing.

He’d said things along the lines of Bombelli being old and senile, a detriment to the younger generation, and that he’d nearly ruined the century-old reputation of *Annals*.

But Tao Zhexuan was a man of decorum; he couldn’t possibly repeat those exact words.

”Reconsider...”

Li Dong’s expression was calm, but internally he was already starting to grumble.

’Seriously? You could at least offer some real perks!’

’You’re all the same! First *Inventiones Mathematicae*, then *Acta Mathematica*, and now you!’

’All you guys ever offer is an expedited review or a special seminar!’

’Can’t any of you talk about something more tangible?’

’Are you top journals broke? Just throw some money at me! What if I’m not strong enough to resist?’

Li Dong sighed to himself.

These top-tier mathematics journals never paid authors for their papers, let alone offered any kind of bonus.

Everyone was in it for the prestige and academic honor.

There seemed to be an unwritten rule throughout the entire world of pure mathematics.

Getting a paper accepted by *Annals* was considered the greatest reward in itself.

Who would have the nerve to ask them for money?

Well... Li Dong, for one, actually did.

But he hadn’t really thought it through yet, so naturally, he didn’t voice his true thoughts.

”Professor Tao, I’m very grateful for Professor Katz’s kind offer.”

”I’ll need to think about it some more.”

Tao Zhexuan wasn’t surprised that Li Dong didn’t agree immediately.

He nodded.

After all, it was normal for a young man to have a bit of an ego.

He, Tao Zhexuan, had been impossibly proud in his own youth.

Back then, countless titans of the academic world had wanted him as a co-author, and he’d ignored every single one.

He saw in the young man before him a faint trace of the same solitary pride he once had.

’Yes! This boy is just like I was!’

”Alright, then. Take your time to think it over.”

Tao Zhexuan stood up.

”I won’t disturb you any further.”

Li Dong saw him to the door.

As he was stepping out of the room, Tao Zhexuan suddenly turned back.

”Mr. Li Dong.”

”Yes?”

”That paper... it’s truly beautiful.”

With that, Tao Zhexuan turned and walked down the hallway.

Li Dong closed the door and let out a long sigh.

”I really need to talk to Old Liu about this.”

Li Dong pulled out his phone again, preparing to give Liu Ruochuan another call.

However, before he could dial the number, his phone vibrated.

It was a WeChat message.

He instinctively tapped it open.

[Bernhard Riemann]: @AcademicGrind