After taking the advice, I became a top student.

Happy times are always fleeting.

After dinner, everyone gathered at the hotel and returned to their respective schools that same day.

While Chen Mo was still on the train returning home, Zhao Yiming, far away at Yanbei University in Beijing, had already received a scanned copy of the exam paper sent from Chengdu.

With today's technology, marking papers no longer requires going to the site, and the double blind marking method also largely eliminates the possibility of cheating.

Zhao Yiming yawned.

He felt that grading these students' exam papers was more tiring than doing his own research. These kids were academic superstars in front of outsiders, but when it came to exams, they would do some sneaky things.

For example, adding self-created theorems or formulas during the proof process to force their derivation to be valid.

He can't afford to be lax in the grading process, otherwise he could easily be fooled.

"what?"

"This little guy is pretty good!"

"Clear thinking, complete steps, and correct conclusion!"

Zhao Yiming nodded secretly. He had corrected so many test papers, and this was the first one he had seen that had completed the third question completely.

Unfortunately, the scanned copies they received were all encrypted, so he couldn't see who the owner of the exam paper was.

After carefully examining the proof once more, he clicked the mouse to move on to the next exam paper.

Shaking my head, shaking my head...

Zhao Yiming marked one test paper after another with an X.

To be honest, the number theory questions in this year's provincial competition were a bit too difficult.

It's even more difficult than the provincial competition, and even more difficult than the CMO. Very few candidates in Chengdu this year were able to solve it.

The main issue was that the knowledge points tested were quite advanced, and many math majors might not even be able to grasp them well.

"Um?"

Suddenly, Zhao Yiming looked at the exam paper in front of him and paused for a moment.

The third question on the test paper in front of him was also filled with the solution process, but the solution to the third sub-question was one he had never seen before.

He had seen many exam papers like this today, but unlike the fabricated proofs he had seen before, this little guy got the first two questions right.

Composing himself, Zhao Yiming carefully deduced the proof process.

"First, transform the original expression... then extract the features and..."

Zhao Yiming quickly took out a draft paper and pen from the side and began to deduce.

Five minutes passed, ten minutes passed...

"You've got something special."

Zhao Yiming looked at his draft paper and found that he couldn't find any errors in the proof process.

In other words, this solution method is also correct?!

However, Zhao Yiming soon discovered the trick.

∑_{k=1}^qχ(k)e^{πik/q}=√q⋅ε(χ).

"Where does this equality come from?"

"You little rascal, you've smuggled in your own agenda again!"

Zhao Yiming was both amused and exasperated. He had spent so much time deducing the reasoning, and he had almost been fooled by this kid.

It must be admitted that this little guy is quite capable. His proof process was flawless, and in fact, the equality he added to the proof is likely true.

Even the proof process itself was flawless. He actually quite admired this method of proof; it was concise, elegant, and possessed a unique mathematical beauty that even pleased this old man who had dedicated his life to mathematics.

However, this little guy lacked the proof of this equality, so I'm sorry, points will definitely be deducted for the steps.

The last sub-question of the third question was worth 15 points. After thinking about it, Zhao Yiming decided to deduct 5 points for the steps.

Only after completing these steps did he click the mouse to continue grading the next exam paper.

Although the exam paper in front of him had changed, his mind was still filled with the proof process on the previous exam paper.

"This kid really has a knack for math."

Zhao Yiming was somewhat tempted.

He could sense the little guy's intelligence just from the proof he had just witnessed.

In China, there are many students who are good at solving math problems, but very few of them end up becoming true mathematicians.

The difference between the two lies in this touch of spirituality.

Or rather, this is talent!

After grading the next exam paper, Zhao Yiming, who had been thinking about it, returned to the previous exam paper and began to study the unique proof carefully once again.

Ok?

Suddenly, Zhao Yiming's eyes widened as he looked at the equation on the test paper.

∑_{k=1}^qχ(k)e^{πik/q}=√q⋅ε(χ).

Like a thunderclap in his mind, like a lightning bolt cleaving through the darkness, "Isn't this exactly what Old Zheng needs?"

Zheng Mingyang needs to connect S(χ)=∑_{n=1}^∞χ(n)/ne^{−n/q²} with a certain finite sum, and the equality on this exam paper shows that the exponent and ∑χ(k)e^{πik/q} can be calculated precisely.

This equality is exactly the bridge that Lao Zheng needs, as it connects a finite exponent with a number-theoretic function.

If Lao Zheng can use some kind of Poisson summation or duality to transform the infinite sum into a finite exponential sum, then the proof of Lemma 3.2 in his paper can be completed!

Zhao Yiming stared blankly at the equation, his mind in turmoil.

He never imagined that the problem that had stumped the two of them, mathematics professors, for their entire lives, would be solved so easily by a little kid. It was simply unbelievable.

Of course, he also understood that the little guy probably just had a sudden inspiration while doing the problem and did not yet have the ability to solve this kind of problem.

He took out his phone and dialed Zheng Mingyang's number. "Old Zheng, come over here right now! I know how to prove Lemma 3.2!"

"What?"

Zheng Mingyang, who was sitting at his desk in a state of ecstasy, suddenly sat up with a start. He jumped up and rushed out of the office with a more agile figure than a young man. Before he could even lock the door, he had already disappeared.

Zhao Yiming's office

"Slow down, slow down."

Zhao Yiming's heart tightened; he was genuinely afraid that his old friend might trip and fall due to excessive excitement.

At this age, falling down is no joke.

"How do you prove it?"

Zheng Mingyang, panting heavily, approached Zhao Yiming's desk and asked impatiently.

"Take a look at this question."

Zhao Yiming rotated the computer screen so that Zheng Mingyang could see it.

"Isn't this the problem I gave to the Chengdu Provincial Mathematical Society?"

Zheng Mingyang frowned, wondering why his old friend wanted him to look at high school students' questions.

"Look at his solution."

Zheng Mingyang was a veteran professor. Although he was extremely excited, he quickly controlled his emotions and did not ask any more questions. He held his breath and focused on the exam paper that was about to be reviewed.

"Please sit down."

Zhao Yiming found a chair, handed over some draft paper and a pen, knowing this was going to be a long battle.

However, before Zhao Yiming could even sit down, Zheng Mingyang slammed his hand on the table. "Yes, that's it! How come I didn't think of using the exact value of the quadratic Gaussian sum to construct a finite approximation?!"

"In the traditional method, the integral expression of S(χ) introduces a kernel function e^{-t}. If this kernel function can be approximated by the finite sum of the exponential sums, the infinite sum can be truncated."

"But here, the Gaussian sum modulo q has an exact analytical expression!"

"so……"

Zheng Mingyang had been studying this problem for a long time, and the moment he saw Chen Mo's equation, he suddenly understood.