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My Study Chat Group is Full of Real Big Shots

Chapter 327 - 284: Mathematics, There Has Always Been Only One Mathematics (2) · Nov 03, 2026

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Chapter List (387)

Page number / 8

301 Chapter 301 - 273: The Fish Took the Bait 2026-11-03
302 Chapter 302 - 274: Mr. Yang 2026-11-03
303 Chapter 303 - 274: Mr. Yang 2026-11-03
304 Chapter 304 - 274: Mr. Yang 2026-11-03
305 Chapter 305 - 274: Mr. Yang 2026-11-03
306 Chapter 306 - 275: I Call the Shots in This Group 2026-11-03
307 Chapter 307 - 276: Human Dominoes 2026-11-03
308 Chapter 308 - 277: In Need of an Anchor 2026-11-03
309 Chapter 309 - 277: Needing an Anchor 2026-11-03
310 Chapter 310 - 278: I’ll Explain More Problems to You Tonight 2026-11-03
311 Chapter 311 - 279: Hello, Master 2026-11-03
312 Chapter 312 - 279: Hello, Master (2) 2026-11-03
313 Chapter 313 - 279: Hello, Master (3) 2026-11-03
314 Chapter 314 - 279: Hello, Master (4) 2026-11-03
315 Chapter 315 - 280: Wow—So Much Food 2026-11-03
316 Chapter 316 - 280: Wow—So Much Food 2026-11-03
317 Chapter 317 - 280: Wow—So Much to Eat 2026-11-03
318 Chapter 318 - 280: Wow—So Much Food 2026-11-03
319 Chapter 319 - 281: We Can Make This Needle 2026-11-03
320 Chapter 320 - 281: We Can Make This Needle 2026-11-03
321 Chapter 321 - 281: We Can Make This Needle 2026-11-03
322 Chapter 322 - 281: We Can Make This Needle 2026-11-03
323 Chapter 323 - 283: Fudan University Open Class 2026-11-03
324 Chapter 324 - 283: Fudan University Public Lecture 2 2026-11-03
325 Chapter 325 - 283: Public Lecture at Fudan University (3) 2026-11-03
326 Chapter 326 - 284: Mathematics, There Is Only One Mathematics 2026-11-03
327 Chapter 327 - 284: Mathematics, There Has Always Been Only One Mathematics (2) 2026-11-03
328 Chapter 328 - 284: There Has Only Ever Been One Math 2026-11-03
329 Chapter 329 - 285: Someone Got Ahead of Us 2026-11-03
330 Chapter 330 - 285: Someone Has Gone Ahead of Us 2026-11-03
331 Chapter 331 - 285: Someone Has Gotten Ahead of Us 2026-11-03
332 Chapter 332 - 286: They Are Gambling 2026-11-03
333 Chapter 333 - 286: They Are Gambling 2026-11-03
334 Chapter 334 - 286: They Are Gambling 2026-11-03
335 Chapter 335 - 287: Never Use Petty Cleverness as a Means to Advance Scientific Research 2026-11-03
336 Chapter 336 - 287: Never Use Petty Tricks as a Means to Advance Research 2026-11-03
337 Chapter 337 - 287: Do Not Use ’Petty Cleverness’ as a Means for Scientific Advancement 2026-11-03
338 Chapter 338 - 288: Got You 2026-11-03
339 Chapter 339 - 288: Got You 2026-11-03
340 Chapter 340 - 288: Caught You 2026-11-03
341 Chapter 341: Li Criterion 2026-11-03
342 Chapter 342 - 289: Li Criterion 2026-11-03
343 Chapter 343 - 289: Li Criterion 2026-11-03
344 Chapter 344 - 290: Joint Training 2026-11-03
345 Chapter 345 - 290: Joint Training 2026-11-03
346 Chapter 346 - 290: Joint Training 2026-11-03
347 Chapter 347 - 291: An Open Invitation 2026-11-03
348 Chapter 348 - 291: Open Call for Papers 2026-11-03
349 Chapter 349 - 291: Public Call for Submissions 2026-11-03
350 Chapter 350 - 292: Your Journal’s Review Process Lacks Rigor, I Decline to Submit 2026-11-03

Chapter 327: Chapter 284: Mathematics, There Has Always Been Only One Mathematics (2)

”...This is as far as I can go.”

She sat down.

From the stage, Li Dong let out a thoughtful hum.

”This student here is already standing at the door to the next stage.”

He nodded at her.

”Your approach is correct.”

”It just doesn’t lead to the answer for this particular problem.”

”It leads to another mountain—one that people like Jacobi, Klein, and Hilbert spent their entire lives thinking about.”

”Let’s conquer this mountain today before we talk about the next one.”

The moment he said that,

a few of the graduate students in the audience

suddenly sat bolt upright.

They had figured it out.

What Li Dong called ”the next mountain.”

It was modular forms.

After the three had finished answering,

the classroom, contrary to expectation, grew even quieter.

Logically speaking, these three answers had already covered the ”three most common approaches” to this problem.

One from analytic number theory, one from geometry, and one from modular forms. 𝒻𝑟ℯℯ

’How else could it be answered?’

From the stage, Li Dong looked down at the audience.

A faint smile played on his lips.

”All three answers are correct.”

”But none of them are complete.”

”The first student’s answer tells you for which *n* there is a solution, and for which *n* there isn’t.”

”The second student’s answer tells you that the total number of solutions is approximately πN.”

”And the last student’s answer tells you that this can ultimately be mapped to a generating function.”

”But there’s one thing all three of them failed to answer.”

He paused for a moment.

”For a specific *n*, how many sets of solutions are there exactly?”

”The precise number of sets.”

”What ultimately determines the number of solutions?”

”It’s not the ’yes/no’ from the prime factorization.”

”Nor is it the ’approximation’ from the area estimation.”

”And it’s certainly not the vague description of the object θ².”

”It’s a closed-form formula, precise for every single *n*.”

At this point, everyone in the audience was listening in silence.

Only then did they realize.

The three paths they had just proposed didn’t touch upon this core issue.

There was an awkward feeling of, ’I answered correctly, but it seems I was answering a different question.’

Li Dong turned around and wrote a line on the blackboard.

[r₂(n) = 4·(d₁(n) − d₃(n))]

Then he continued writing on the line below.

[d₁(n) = #{d such that d divides n, d ≡ 1 (mod 4)}]

[d₃(n) = #{d such that d divides n, d ≡ 3 (mod 4)}]

He put down the chalk and turned around.

”This is a precise formula given by Jacobi in 1828.”

”For any given *n*, to find the number of solutions, you count how many of its divisors are congruent to 1 modulo 4 and how many are congruent to 3 modulo 4. Take the difference, then multiply by 4.”

”And you’re done.”

A collective gasp went through the audience.

A few graduate students couldn’t help but mutter in low voices.

”How... How did he come up with that?”

”How are these two things even related?”

Li Dong chuckled.

He raised his hand and wrote a new set of symbols on the blackboard.

[ℤ[i] = {a + bi : a, b ∈ ℤ}]

”Gaussian Integers.”

”a plus b times i.”

”Where a and b are both integers.”

”In this new number system, you do one thing.”

”You take an integer *n* and expand it into its prime factorization within ℤ[i].”

He drew a branching diagram on the blackboard.

”Among ordinary integers, 5 is a prime number.”

”But in ℤ[i], 5 = (2 + i)(2 − i).”

”It splits.”

”What about 3? In ℤ[i], it’s still a prime. It doesn’t split.”

”And 7? It doesn’t split either.”

”Why do some primes split while others don’t?”

He glanced at Zhang Heng.

Zhang Heng’s eyes were already shining, and he answered loudly,

”Those congruent to 1 modulo 4 split.”

”Those congruent to 3 modulo 4 don’t.”

Li Dong nodded.

”So what is r₂(n) actually asking?”

”It’s asking in how many ways you can factor *n* into a product of two Gaussian Integers.”

”This is the ’divisor function’ within the Gaussian Integers.”

”Change the number system, and the problem shifts from ’counting solutions’ to ’counting divisors’.”

”Change your perspective, and the problem transforms from a ’combinatorial problem of algebraic transformations’ into a ’familiar factor-counting problem from number theory’.”

He paused, then pointed to Qiu Jiayuan.

”You there. Your area estimation just now stated that the average order of r₂(n) is π.”

”Do you know why it’s π?”

Qiu Jiayuan said with some hesitation,

”The formula for the area of a circle.”

”Yes.”

Li Dong shook his head.

”And no.”

”The formula for the area of a circle is just a surface-level phenomenon.”

”If you take this solution-counting function r₂(n) and form a Dirichlet series...”

Li Dong wrote another line on the blackboard.

[Σ r₂(n)/n^s = 4·ζ(s)·L(s,χ)]

”Analytically, it splits into the product of two things.”

”One is the Riemann ζ-function.”

”The other is the Dirichlet L-function for the non-trivial character modulo 4.”

”The first term of the product has a pole at s=1, with a residue of 1.”

”The value of the second term of the product at s=1 is π/4.”

”Multiply the two together, and you get that π.”

”That is the real reason why the average order is π.”

”That π didn’t just fall out of a circle.”

”It fell out of an L-function.”

The people in the audience were now completely dumbfounded.

Qiu Jiayuan was left breathless.

His earlier answer, ”the area of a circle,”

was just an accidental consequence of this L-function at s=1.

Li Dong turned around.

”Let’s take another step back.”

”What was that L(s,χ) from just now?”

”It translates that simple parity modulo 4 into an analytic object.”

”This class of objects has a name.”

”It’s called... the very first, the absolute lowest-dimensional example of an automorphic L-function.”

”The automorphic L-function on GL(1).”

Up to this point, he hadn’t mentioned the name ”Langlands” even once.

But when the graduate students present heard this, the color drained from their faces.

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