"A paper?"
Hearing Academician Hu's request, Xiao Yi's eyes lit up. As a high school student, he had never encountered anything like academic papers before.
Well... he had only come across "argumentative essays" in Chinese composition classes.
Yet Academician Hu had actually sent him a paper and asked him to take a look at it. His interest was immediately piqued.
"Alright, I'll take a look."
He then downloaded the paper Academician Hu had sent and began reading.
The first thing he saw was information about the author, someone named Zeng Ang. He was actually a graduate student at USTC's Academy of Mathematics and Systems Science. His advisor was not Hu Guangde, however, but someone else.
Of course, Xiao Yi did not linger over that information. Looking into other people's details was not exactly a good habit, so he turned to the next page.
"There is an inseparable relationship between the convergence of the Riemann Integral and Lebesgue Integrability. First, the Riemann Integral convergence theorem is proved in detail, then... finally, an application is given."
This was the abstract, mainly explaining what the paper covered.
Then came the main text.
Although he had only been studying for a few days, once he began reading the main text, Xiao Yi did not find it difficult at all. He had seen almost everything mentioned in the paper in his books. Even when he encountered something unfamiliar, a brief derivation in his mind was enough for him to understand it completely.
This paper was probably just something this graduate student named Zeng Ang had written for practice. Its depth was not particularly high; at least, there was absolutely no problem with the theoretical derivations in the first half.
Very quickly, Xiao Yi reached the final section of the paper: the application of the conclusions derived earlier.
But after only a few glances, he frowned slightly.
Because he had discovered an error in the application process.
"Using Dirichlet's Test, the convergence of the Riemann Integral (R) cannot be determined directly under the existing conditions."
He could not help wondering whether he had made a mistake. After all, this paper had been written by a graduate student who might one day become his senior. How could it possibly be wrong?
There had to be an explanation or a correction later on.
So he continued reading. Yet after turning another page, he found that the graduate student senior had directly concluded, "Therefore, the conclusion is valid."
Below that was the summary.
"This..."
Xiao Yi found himself in a dilemma.
He went back and read it over again. In the end, he reached the same conclusion: it was wrong!
However, Academician Hu was the one who had asked him to read this paper. Although the student named Zeng Ang was not Academician Hu's advisee, they surely had some connection.
Would it be inappropriate to point out the mistake directly?
But he soon stopped dwelling on it. Mathematics was a science where "right was right, and wrong was wrong."
So he finally spoke. "Academician Hu, I've finished reading it."
Hu Guangde smiled. "Good. Tell me what you think."
"Alright." Xiao Yi nodded. "The first three sections of this paper are correct and contain no errors. The deductions in them should also be relatively basic material in Real Analysis, though the paper does conduct some research to a certain extent."
"However, there is an error in the fourth section, the part concerning the application of the deductions."
"There is an error in the part that uses Dirichlet's Test to determine the convergence of the Riemann Integral (R). Under the currently known conditions, that judgment cannot be made directly."
Hearing Xiao Yi's words, Hu Guangde raised an eyebrow before asking, "Then how do you think this error should be corrected?"
Xiao Yi was taken aback. He had not expected Hu Guangde to ask how he had discovered it rather than how to fix it.
But he answered directly, "I need to start from the beginning."
"Let α∈R, and let f(x)=sin1/x/x^α, x∈(0,1]. Then the following conclusions hold: (i) when α≤0, f∈R[0,1] and f belongs to L[0,1]..."
Xiao Yi began his explanation. On the other side of the video call, Hu Guangde's smile grew increasingly brilliant, while the other professors, including Liu Bin, gradually opened their mouths wider and wider.
This paper was indeed nothing much to professors at the level of master's and doctoral advisors. But the person on the other end was a second-year high school student who had only gone through Real Analysis once, and had not even spent two full days on it!
He had studied it for less than two days, yet he had already mastered Real Analysis to such a degree that it normally took ten readings to grasp. He had even pointed out an error in a paper written by a graduate student after reading it?
That graduate student named Zeng Ang seemed to be the student of Academician Hu's student, didn't he?
In other words, Hu Guangde's grandstudent. He certainly had ability.
After their amazement subsided, the professors exchanged glances. Finally, all of them fixed their eyes on Liu Bin.
Old Liu, old Liu, where on earth had he found a student like this?
Liu Bin wore an even more sorrowful expression.
Before, he had wanted to bring Xiao Yi to their school and properly temper the arrogance of the other geniuses in the Gifted Class. But he had never imagined that Xiao Yi would also temper the arrogance of their mathematics professors along the way.
His confidence was about to disappear.
He thought again of the forty books he had given Xiao Yi. With this guy's reading efficiency, could so many... so few books last him a month?
At most, difficult books such as Algebraic Geometry and Algebraic Topology, along with those english textbooks, might make Xiao Yi spend a little more time.
Really...
Such a despairing gap.
"...Make the substitution x=... in the integral ∫2πk+π,2πk |sint/t|dt. We can easily prove the validity of (16), thereby obtaining f∈L[0,1]. Finally, by the Cauchy Criterion, (iv) is proved."
"Proof complete."
Xiao Yi's voice came through the phone, signifying that he had already corrected the error in the paper.
"Very good."
At last, Hu Guangde nodded. He even raised a thumbs-up. If Xiao Yi had been standing in front of him, he might have applauded outright.
"Very well explained. This was precisely the main reason I asked you to read this paper."
"The error in this paper is not especially obvious, because the discussion of the Riemann Integral and Lebesgue Integrability in section 3.2 creates the illusion that the originally missing condition has been included, when in fact it has not."
"Such an error may be difficult to guard against even for experienced people."
"Only those who have truly mastered this part of Real Analysis can avoid such problems."
"Furthermore, the Cauchy Criterion you used at the end made this section even more impressive."
This Academician of Mathematics gave his final assessment. "In short, I'm already looking forward to seeing you at the Academy of Mathematics and Systems Science at USTC."
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