Chapter 184 - 179: Do Me a Favor
Tian Gang’s offer of help left Li Dong utterly confused.
What kind of help could he possibly offer an academician?
"Teacher Tian, are you pulling my leg?" Li Dong chuckled. "What could I possibly help you with?"
Tian Gang waved his hand, his expression turning serious.
"No, no, I’m serious."
"I have several doctoral students, and their respective research directions overlap with the correspondence between zero-pair correlation and local ramification indices that you proposed in your paper on Montgomery’s Conjecture on Association."
"Your paper not only extended the boundary of Montgomery’s theorem from |α|
"It’s no longer just a theorem in number theory."
"It’s a tool."
Tian Gang looked at Li Dong and said slowly.
"A foundational tool that can be repeatedly cited and extended throughout the entire fields of the Langlands Program and analytic number theory."
Li Dong was stunned for a moment, then he understood what Tian Gang meant.
The value of a truly significant mathematics paper lies not only in the problem it solves.
It lies even more in how many new foundations and directions it can provide for subsequent research.
This is a common occurrence in the history of mathematics.
The most classic example is Riemann’s 1859 eight-page paper, "On the Number of Primes Less Than a Given Magnitude."
It only provided a conjectured expression for the prime-counting function, yet it introduced the core concepts of the Riemann zeta function and its non-trivial zeros, directly giving birth to the entire modern framework of analytic number theory.
More than one hundred and sixty years have passed, and number theorists worldwide are still using that paper as a foundation to build one academic edifice after another.
Or take Andrew Wiles’s 1995 paper proving Fermat’s Last Theorem.
It solved a 358-year-old conjecture of the century.
Moreover, in the process of the proof, it developed modularity lifting techniques and deformation ring theory to an incredible degree.
In the subsequent twenty-plus years, it has been widely used in various branches of the Langlands Program, directly spawning at least a hundred top-tier follow-up papers.
And in Tian Gang’s view, Li Dong’s Montgomery paper possessed this same kind of "foundational" academic value.
Tian Gang continued.
"I’m advising a doctoral student."
"He’s been working on a project recently, focusing on using the statistical properties of the pair correlation function to study the fine lower bounds of the zero-spacing distribution of Dirichlet L-functions."
"Previously, the academic community only had rigorous results for the interval |α|
"But after your paper came out, he suddenly saw a complete path to solving his problem."
Tian Gang smiled.
"But just having a path isn’t enough. He’s encountered problems in the specific derivations."
"So I was hoping you could give him some guidance."
Li Dong finally understood.
’So that’s what this is about.’
To be honest, he didn’t think it was a big deal.
Other people using the results of his paper as a tool to conduct new research, write new papers, and earn new honors...
This wasn’t a loss for him at all; on the contrary, it was a good thing.
Academic influence is measured by citation count.
The more your paper is cited, the higher the standing of your work in the academic world.
And what truly determines a mathematician’s lifelong academic status isn’t a single breakthrough in a particular year, but the impact of their work on the entire field.
This is also why some awards in mathematics have a completely different selection logic than the Fields Medal.
For example, the Wolf Prize in Mathematics and the Abel Prize.
The Fields Medal places more emphasis on breakthroughs.
The Wolf Prize and the Abel Prize, on the other hand, recognize a mathematician’s lifetime of academic contributions.
In a sense, these two lifetime achievement awards are a better representation of a mathematician’s ultimate place in the history of mathematics than the Fields Medal.
Historically, the scholars who have received these two awards are almost all figures who are considered founders of their respective fields.
For example...
Chen Shengshen!
So, what Tian Gang was doing today wasn’t entirely for his own doctoral student.
He was playing a much bigger game.
Everyone had their eyes on the Fields Medal.
Including Li Dong himself, who had declared in front of CTV’s cameras that he would win the Fields Medal before he turned thirty.
But Tian Gang saw something beyond the Fields Medal.
Li Dong was a part of Yan University.
The results he produced should rightfully be embraced, amplified, and carried forward by Yan University’s academic ecosystem.
If Li Dong’s paper became a new research cornerstone for the Langlands Program, and all the subsequent results built upon this cornerstone came from other places, or even other countries...
Then he, Tian Gang, couldn’t afford to lose face, and neither could Yan University!
Tian Gang would not allow such a thing to happen.
"No problem at all."
Li Dong agreed readily.
"Just ask me anything you need to."
Tian Gang nodded, then turned and called out to the audience.
"Zhe Yuan, come up."
A young man in the second row of the lecture hall, near the aisle, stood up.
He looked to be twenty-seven or twenty-eight, and he was clearly nervous as he walked up to the stage.
Song Zhe Yuan was a third-year doctoral student under Tian Gang, specializing in analytic number theory. He had previously published two decent papers, but they were in Q2 journals, still a far cry from being truly impressive.
When Song Zhe Yuan stood before Li Dong, he didn’t feel awkward at all about Li Dong being younger than him.
’Are you kidding me?’
’Someone who can be the first author on a paper in the *Mathematical Annals* and propose a brand-new criterion for the Langlands Program? Forget being a few years younger—even if he were a high school student, I would still have to seek his guidance with the utmost seriousness.’
"Teacher Li Dong, hello."
Li Dong quickly waved his hands.
"Senior, don’t call me Teacher. Just call me by my name."
Tian Gang, standing to the side, said with a smile.
"Just go ahead and ask."
Song Zhe Yuan quickly opened his notebook.
"My current research direction is the fine statistics of zero-spacing for Dirichlet L-functions."
"Specifically, I want to use the asymptotic behavior of the pair correlation function in the extended interval from your paper, combined with the correspondence between the ramification index and convergence interval you proposed, to characterize the lower bound for the minimum spacing of non-trivial zeros of a general Dirichlet L-function modulo q."
"Your paper proved that for |α|∈[0, 4], F_T(α) is in perfect agreement with the GUE prediction, and it also provides interval-matching rules for different conductor characters. I can use this conclusion directly as a core lemma."
"But..."
He pointed to a complex summation formula in his notebook.
"When I try to extend the pair correlation conclusion for the ζ-function to the family of a general Dirichlet L-function by analogy, the orthogonality relations of the character sum introduce extra cross-terms."
The moment he said this, Li Dong knew what his problem was.
’It’s just that...’
’These cross-terms can be bounded by the large sieve inequality when |α|
Li Dong nodded to himself. ’Teacher Tian’s student is definitely skilled.’
’The question is very precise.’
So he said.
"The core of your problem isn’t actually the large sieve, but the way you’re matching the intervals."
Li Dong wrote a line of formula on the blackboard.
"When you replace the ζ-function with a general L(s, χ), an additional summation over the character χ appears in the definition of the pair correlation function."
"The remainder term of this summation isn’t a fixed constant; it’s directly tied to the conductor of the character χ and the length of the corresponding convergence interval."
"The reason you feel the remainder term can’t be bounded is that you’re applying the full interval [0, 4] uniformly to the characters of all conductors, without performing a stratified matching."
He drew a line on the blackboard, stratifying the conductors by the number of their prime factors.
"The correct approach is to stratify the characters according to the size of their conductors, match the characters in each stratum with the corresponding convergence interval length from the paper, use the large sieve to estimate the remainder term within each stratum separately, and then take a weighted sum of the estimates from all strata."
"If you do that, you’ll find that the contribution from the cross-terms is controllable in each stratum, and the final total remainder term will still be o(1)."
Song Zhe Yuan’s eyes lit up instantly.
"Stratify and match intervals by conductor!"
He quickly jotted it down in his notebook.
"Yes, yes... I was trying to apply the full interval to everything, completely ignoring the correspondence between the conductor and the interval. This way, the modulus for each stratum is fixed, making the orthogonality relations much cleaner to use!"
His train of thought was completely unblocked.
"Then, within this stratified matching framework, can the higher-order contributions from squares of primes and cubes of primes also be handled with a similar method?"
Li Dong nodded.
"Yes, but the contributions from higher powers of primes decay more rapidly, so the corresponding intervals can be made tighter, and the stratification can be coarser."
"Specifically..."
He continued his derivation on the blackboard.
At this moment, the expressions on the faces of the graduate students auditing the discussion began to grow subtle.
Most of them didn’t specialize in analytic number theory or the Langlands Program, and they were hardly experts on the technical details of Montgomery’s Conjecture on Association.
But strangely, when Li Dong was answering Song Zhe Yuan’s questions, they could actually understand the core Logic behind the character stratification, interval matching, zero criteria, and local-global compatibility.
The feeling was very peculiar.
It was as if every sentence Li Dong spoke came with some kind of built-in translation function.
It was clearly a top-tier academic discussion at the Doctoral Level, yet the master’s students present, and even a few upper-level undergraduates who had come to listen in with their advisors, could faintly grasp the outline of this frontier field.
Meanwhile, Tian Gang, sitting to one side of the stage, frowned slightly.
’Something’s not right.’
He looked at Song Zhe Yuan.
He had been advising this student for three years; he knew his capabilities well.
He was smart and hardworking, but in terms of academic ability and on-the-spot thinking, he was only upper-middle among the doctoral students he had mentored.
During their usual project discussions, Song Zhe Yuan would often go off track on key issues, requiring Tian Gang to pull him back.
’But what’s going on today?’
From the moment he stood up until now, every question Song Zhe Yuan had asked was perfectly on point.
’This doesn’t seem like the usual Song Zhe Yuan.’
Tian Gang looked at Li Dong again.
’Is he the one guiding him?’