"...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.
’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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