Chapter 180: I Might Need to Borrow the Blackboard

After a 30-minute lunch break, at 2 p. m., Xu Qingzhou returned to the lecture hall and was surprised to encounter Keshalin again.

This lecture was even more popular than the morning one; he had arrived 30 minutes early, yet barely managed to secure a seat.

"Too many people," Xu Xu Qingzhou remarked.

Keshalin sat beside him, smiling faintly and nodding. "This is Professor Maynard's lecture."

Professor Maynard currently teaches at Oxford University and is a leading figure in the field of prime numbers; the topic of this lecture is the Riemann Hypothesis.

"What do you think the Riemann Hypothesis is?" Keshalin asked.

Xu Xu Qingzhou thought for a moment and said, "To us, the Riemann Hypothesis might be similar to Fermat's Last Theorem before algebraic geometry emerged."

The Indian guy smiled, glancing challengingly at Xu Xu Qingzhou.

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Keshalin asked Qingzhou: "Do you think he can succeed?"

After all, it's common for renowned scholars to announce they've proven a conjecture, only to have it refuted the next day.

The Indian guy spoke in curry-flavored English:

In the audience, Xu Xu Qingzhou opened his notebook and began listening intently.

Yet some were interested; one young man pulled out his business card: "Sir, I'm from Loughborough University. Would you be interested in collaborating on this topic?"

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"Here, we define $\pi_2(x)$ as the number of twin prime pairs less than or equal to $x$. That is, if there exists a prime $p$ such that both $p$ and $p+2$ are prime, then the Twin Prime Conjecture is equivalent to $\lim_{{x\to\infty}}\pi_2(x)=\infty$."

Several people wanted to ask questions—about four or five, including Xu Xu Qingzhou, who had been deep in thought, and the Indian guy.

Lecture content: New large value estimates for Dirichlet polynomials.

Xu Xu Qingzhou shrugged and returned to his seat, abandoning the pointless argument; he thought, if it were this simple, he'd have solved it during the New Year holidays.

Soon, a circle of people gathered around, naturally including Qingzhou and Keshalin.

The proof still relied on Fourier analysis.

Keshalin chuckled: "I also think it won't work."

Watching the man's confident demeanor, Xu Xu Qingzhou couldn't help saying, "Now, for the remaining $m$, with respect to $s - s/2 - s/2 - s$, it must satisfy $r - 2 \leq w(m) \leq r$, but clearly, continuing the calculation will produce a result contradicting this condition."

Beside him, the uninvited Indian guy felt a twinge of envy—again this annoying guy. Hmm, he'd like to see what brilliant insight he could offer.

No choice—Professor Maynard is among the world's top mathematicians, and the topic, the Riemann Hypothesis, is currently extremely hot. Soon, a middle-aged man in a suit entered the lecture hall.

Next door, two people's conversation had shifted from the Riemann Hypothesis to the Twin Prime Conjecture; one had even laid out several sheets of paper and was scribbling on them.

"No." This time it was Xu Xu Qingzhou who spoke with certainty.

But everyone here had some substance; no one was overly excited, preferring to proceed cautiously.

"Very accurate," Keshalin's eyes lit up, fully agreeing with Qingzhou's remark.

By this point, the experts in the front rows had already opened their notebooks and begun calculating along with Professor Maynard's reasoning.

Or like the Eiffel Tower's blueprints appearing in the Stone Age.

Professor Maynard likewise got straight to the point; the large screen displayed pre-prepared content.

"Correct."

Keshalin stared intently at the paper, thinking deeply.

Professor Maynard looked at Xu Xu Qingzhou and said, "Sir, you may speak." It was significant.

Someone raised a hand.

Professor Maynard scanned the lecture hall and said, "I'm sorry, but due to time constraints, I can only answer the last question."

"Of course, we must mention Mr. Zhang Zhang Yitang's result on twin primes; by supplementing and improving this result, we found we could recalculate the new large values of Dirichlet polynomials."

Professor Ajil was somewhat well-known in number theory circles; upon hearing his name, the skeptical glances around diminished.

One hour passed.

Professor Maynard's achievement was like extending this bridge by another segment—though not fully repaired, it now allowed some exchange in economy, transportation, culture, and more.

Unlike the others, Xu Xu Qingzhou felt disappointed; this method was terrible. If they continued like this, they wouldn't even come close to Zhang Zhang Yitang's prime methods—it was a dead end.

Solving a mathematical problem is like building a bridge: you build a bit on the left, a bit on the right, but a section in the middle is missing.

Keshalin's beautiful eyes regarded Qingzhou curiously.

"During this period, we made the first substantial improvement to Ingham's classical bound on the zeros of the Riemann zeta function from around 1940."

He added, as if to validate his conclusion: "My teacher, Professor Ajil, also endorses this method."

For example, raising a key phase matrix to the sixth power allows better description and analysis of the function's behavior across different scales.

The Indian guy fell silent but stuck to his view: "No, our calculations are not flawed; if we continue, we will surely reach a result."

The main conclusion was presented by an Indian guy, who used a modified weighted sieve method, similar to Zhang Zhang Yitang's, adjusting the distribution of arithmetic progressions.

Professor Maynard's gaze landed on Xu Xu Qingzhou; compared to others, this questioner was far too young.

"But you've already counted $m$ twice in $s$; your lower bound is inaccurate," Xu Xu Qingzhou said with a smile.

At the 30-minute mark of the lecture, a turning point—or rather, a clever choice—emerged.

Compared to the Riemann Hypothesis, the Twin Prime Conjecture seemed less out of reach.

In Xu Xu Qingzhou's view, the first few steps were standard and not difficult; judging from the expressions of other attendees, they likely shared his opinion.

He looked forward to this young man raising a novel question.

Xu Xu Qingzhou nodded and said, "Logically, this method of interval improvement requires special techniques to yield positive results."

The question-and-answer session was more than halfway over.

Those around him fell into thought, pondering how to proceed if following the Indian guy's approach.

As a prime number researcher, he immediately thought of the young man who had just proven Cramér's conjecture.

Professor Maynard nodded slightly; this young man had not disappointed him—he spotted the flaw at once. He admitted, "Regrettably, before this lecture, we had only calculated up to this point. The next step is crucial, but we currently have no concrete direction."

"Believe me, if we continue calculating, there's an 80% chance we can prove the Twin Prime Conjecture!"

The Indian guy shook his head: "No, such a situation will never arise—we've already established the lower bound for $s$."

By now, every seat in the lecture hall was filled, and people stood even in the aisles.

Xu Xu Qingzhou nodded, glanced at his notebook, and said, "I have some ideas. If possible, I'd like to borrow the blackboard."

"Please, feel free to use it," Professor Maynard said seriously, gesturing with a hand.

Xu Xu Qingzhou rose calmly and walked toward the podium, where a staff member had already pushed forward a clean blackboard.

Onstage, he picked up a piece of chalk and wrote a complex formula on the board.

$\sum_{\infty}\left(\frac{\gamma\left(\frac{s+\delta}{2}\right)$

(End of chapter)