After taking the advice, I became a top student.

Zheng Mingyang plopped down, picked up a pen and draft paper, and began to deduce.

Rewriting e^{-n/q²} in integral form as e^{-n/q²}=1/π∫−∞∞e^{−t²q²+2πint}dt, and then exchanging the summation and integration, we get S(χ) as the product of the kernel function and the Gaussian sum.

Zheng Mingyang wrote down a series of derivation formulas on the draft paper with lightning speed. He felt the same pleasure as someone who had been holding in their pee for more than ten hours and had finally found a toilet to relieve themselves.

Inspiration flows like water, cascading down endlessly.

The inner summation is a Gaussian sum modulo q, which can be precisely calculated using a method similar to the equality we just saw in the exam paper.

Finally, we obtain |S(χ)|≥c/logq, and the constant c is effective!

Hahahaha...

Zheng Mingyang looked at the formulas he had drawn and burst into laughter.

The problem that had been bothering him for almost half a month was solved in just half an hour, and the entire proof process only took two pages of draft paper.

"genius!"

"This child must be a genius!"

Zheng Mingyang looked up again at the answer on Zhao Yiming's computer screen, his eyes filled with amazement, as if he were looking at a work of art.

"Congratulations!"

Zhao Yiming smiled happily, genuinely pleased for his old friend. He took his friend's draft paper and began to study it carefully, wanting to witness this historic moment.

Zheng Mingyang looked at his old friend and sighed, "I've been bombarding this lemma with the cannons of analytic number theory, only to find that all I need is a small knife, elementary Gaussian sums and estimates."

This child's thinking was so ingenious. He made me realize that some problems don't require complex integrals; they only need to go back to the most basic finite sums.

"Do you know who this little guy is?"

Zheng Mingyang's heart stirred, and he had the idea of ​​taking on a disciple. He had seen this test paper. To get full marks in the second round of the provincial competition and to write such a brilliant stroke showed that this kid had a solid foundation in mathematics and also possessed the most important talent for doing mathematics.

If we don't act sooner, he might never get a chance.

Zhao Yiming shook his head, his gaze still fixed on Zheng Mingyang's draft paper. "You know the rules: we can't know who answered the questions until the final results are released."

"Then we'll just have to wait a bit longer."

Zheng Mingyang was very disappointed.

Suddenly, Zhao Yiming frowned slightly. "Old Zheng, this equation doesn't seem to have a proof."

If it were just a provincial competition question, he could turn a blind eye and just deduct some points for the steps.

However, in mathematical papers, an unproven condition can easily lead to an incorrect final result, and even if the authors are extremely confident, the reviewers will not listen to their explanations.

Zheng Mingyang's smile froze instantly.

He was so excited that he overlooked such an important matter.

But it doesn't matter.

Proving an expression written by a high school student is certainly easier than proving the missing lemma.

Zheng Mingyang picked up his pen again and buried himself in the draft paper to deduce the solution.

……

Participating in the math competition didn't have much of an impact on Chen Mo. After returning to school, he continued to study math and physics every day, supplemented by other subjects, and followed the suggestions from netizens, absorbing knowledge like a sponge.

Because the provincial competition results wouldn't be released for a few more days, the students' discussions were limited to whether Chen Mo could win an award this year. Guo Chao even secretly started betting on it in class.

The wager is that it's your turn to do your duty.

If Chen Mo can get the provincial third place, the odds are 1:1.1. If Chen Mo gets the provincial third place, everyone will place bets for ten days, and he will help others take turns for one day.

The odds for the second-highest province prize are 1:2. If Chen Mo gets the second-highest province prize, Guo Chao will have to help others with their betting for a day.

The odds for the provincial first prize are 1:10.

"Hey kid, these odds are pretty good."

Li Ze clicked his tongue in amazement, "Aren't you afraid that if Chen Mo really wins the provincial first prize, you'll lose everything?"

"What do you know?"

Guo Chao said calmly, "Whether I believe Lao Chen or not is not important. My classmates certainly don't believe it. How likely is it that someone can win first prize in their first provincial competition?"

"Besides, everyone knows what Old Chen's situation was before."

"So everyone will definitely bet on the second or third prize of the provincial prize!"

"As for the odds for first place in the province, they're just there to attract people, understand?"

Li Ze was speechless. He had to admit that Guo Chao was bad at studying, but he was quite good at doing business. That was his talent.

"Why are you only offering savings of one, two, and three? You're playing this trick, aren't you?"

A classmate named Wang Hai approached Guo Chao, looking at him with an all-knowing gaze, and said, "If Chen Mo doesn't win the award, wouldn't you be the bookmaker taking all the profits?"

A single word awakened him from his dream.

The students immediately realized what was happening.

Good grief, Guo Chao is playing mind games with them.

"rnm, refund my money!"

The students immediately protested.

"Don't worry, everyone. You can still bet even if you don't win the prize."

Guo Chao's little mind worked quickly, "But the odds are only 100:1."

"I'm betting for 100 days, Chen Mo won't win the prize."

Wang Hai placed a large bet without hesitation.

"I'll also put it on hold for 10 days."

"I'll also put it on hold for 10 days..."

The students followed suit, some even placing large bets, hoping to recoup their previous bets on provincial duty days and earn even more days.

Guo Chao's forehead began to sweat.

"Brother Hai is awesome!"

"Brother Hai, once the results come out, I'll give you 0.1 days of the days I earned. You deserve it!"

Wang Hai was immediately hailed as a hero by his classmates.

Everyone walked away with smiles on their faces, waiting for the day when the provincial competition results would be announced.

It is foreseeable that that will be a day of harvest.

If this were in a novel, it would at least be a chapter titled "Wang Hai Cleverly Uses a Series of Schemes to Outwit a Shady Merchant and Show His Heroic Spirit."

"What's the point of betting 100:1? I'm familiar with this kind of betting. I'm betting that Chen Mo will win the provincial championship."

Li Ze, refusing to believe it, declared, "I'll bet for 10 days!"

The students looked at Li Ze as if he were an idiot. In their eyes, even if Chen Mo could win an award, he would at most be a provincial third-place winner, maybe even a provincial first-place winner. He simply loved doing chores and cleaning.

Guo Chao was almost moved to tears. "Brother Ze, from now on you're my half-brother!"

In his view, Li Ze's decision to top the provincial rankings was because he saw that he was under too much pressure and was helping him share the burden of his debts. That's what a true brother does.

He subconsciously glanced at Huang Qinghua beside him, then gestured to Li Ze with his eyes.

The meaning couldn't be clearer: Dude, Brother Ze has already set a precedent, so why not help out your brother?

Huang Qinghua rolled her eyes. "Then I'll hold it for one day too."

Guo Chao pouted, clearly dissatisfied with Huang Qinghua's stinginess, but he was still grateful that Huang was willing to help him.

The farce has ended.

Guo Chao approached Chen Mo and asked, "Brother, tell me the truth, how did you do on the exam this time?"

"It should be alright."

Chen Mo thought back for a moment and realized that he had done all the questions, and the ones he knew how to do should all be correct.

A glimmer of hope appeared in Guo Chao's eyes; he saw a glimmer of hope again. "So, is it possible to get third place in the province?"

Chen Mo shook his head, "Third place in the province probably won't work."

Guo Chao immediately deflated like a punctured balloon. "I knew it."

"I will never gamble again."

"I am mortal with gambling and drugs!"

Guo Chao wailed.

"First place in the province is no problem!" Chen Mo added.

Even with a perfect score, you probably wouldn't get a third-place finish in the provincial competition.

Guo Chao, however, did not believe it at all, thinking that Chen Mo was just joking with him.