Chapter 422

Boston.

In December, it was only 6:30 PM, yet the night curtain outside the window had already completely descended.

"Time to eat."

Xu Qingzhou came out of the kitchen carrying the spare rib soup and called out to the girl wearing pink pajamas on the sofa.

"Oh, coming." Song Yao was lying on the sofa reporting her safety to her family; hearing Xu Qingzhou's words, she shuffled over in her slippers to help bring the dishes.

Three dishes and one soup.

Both were very tired, and they swept the food on the table clean like a whirlwind; even Song Yao finished two large bowls of rice.

"Phew~"

After finishing his meal, Xu Qingzhou leaned back in his chair and let out a long sigh of relief; full and satisfied, he finally regained some of his energy.

He looked at the girl opposite him holding her bowl and drinking soup, sighing in his heart again—men and women are indeed different.

Song Yao had just taken a shower before eating, her hair draped casually over her shoulders, her fair skin showing a rosy glow.

She sipped the soup in small mouthfuls; it was clearly just a bowl of ordinary spare rib soup, yet she managed to drink it with an extraordinary sense of ritual.

A pair of beautiful eyes were clear and transparent, and her slightly curled eyelashes were clearly visible.

"So full~"

Song Yao touched her little belly and said contentedly. In a foreign land, she had been so focused on studying that she had long since thrown things like dieting to the back of her mind, going out to eat delicious food with Kaelier every day.

"Do you want some more?" Xu Qingzhou asked with a smile.

"No more."

Song Yao shook her head and touched her waist, feeling as if she had gained weight, thinking to herself that she must find time to buy a scale.

After the meal, Song Yao took the initiative to take on the task of washing the dishes, letting Xu Qingzhou, who had been doing all the heavy lifting, have a good rest.

Xu Qingzhou poured cat food for Er Miao and was then attracted by Song Yao in the kitchen; he leaned against the kitchen door frame watching her tidy up the bowls and chopsticks. Although the winter pajamas were a bit thick, they still couldn't hide the curvaceous figure of the school beauty Song.

They say you shouldn't pat a tiger's bottom.

Xu Qingzhou suddenly wanted to try, and involuntarily reached out to pinch it.

"Xu Qingzhou!"

Song Yao turned her head, glaring angrily.

"It's 'Xu Qingzhou' again now; just a moment ago you were calling me 'Xu Laoshi'."

Xu Qingzhou teased.

"You just wait." Song Yao gritted her teeth.

"Threatening me now? Is this the legendary 'pulling up your pants and not recognizing the person'?"

The person with the loose mouth was kicked out of the kitchen.

After Song Yao finished attending the academic exchange meeting, she was even busier than before.

It was the same for Xu Qingzhou, doing analysis on the Riemann function.

He actually had other realizations.

He lamented that mathematics is just like physics; the more you come into contact with it, the more you have an inexplicable sense of awe, and he marveled that the big shots of the past were truly impressive.

Regarding the calculation of the zeros of the Riemann function, Riemann himself had once calculated several zeros by hand.

Even today, with advanced computers, numerical calculation remains very difficult; for example, in the approximation formula for calculating the z-function derived by Riemann-Siegel, the derivatives here have to be calculated to the 12th order.

Kyoto.

Just as Chu Jiangfeng entered his study, he received a call from Gu Zhizhong.

"Old Gu, take a look at what I sent you."

"You old fellow, you're already fifty or sixty years old, and you're still this excited, what... Holy cow!"

Cambridge.

Naris Doncic was drinking morning tea, flipping through the arxiv website in boredom.

Looking at papers on arxiv is, simply put, like panning for gold in sewage; besides directly searching for a certain mathematician's homepage, it is very difficult to find a decent paper.

A considerable number of people just have an idea on a whim, and after finishing it, they find it's like a pile of dung, but not entirely dung—it's stupid in just the right way. They can't get it into a journal, but it's a pity to throw it away, so they finally just hang it on arxiv to get some citation counts, and can incidentally add some trouble for peers working in the same direction, or consider it as marking pitfalls.

"A Brief Proof of the Riemann Hypothesis... Garbage."

"Attempting to use basic mathematical knowledge such as algebra to solve the Riemann Hypothesis? Hehe..."

"Dirichlet's L-function."

"Proof of the Landau-Siegel Zero Conjecture... Heh, gar... Wait, hold on."

Naris originally wanted to skip it directly, but out of the corner of his eye, he caught sight of the paper's author: "Author qingzhouxu, is it the one I know?"

Naris was stunned, and with doubt, he clicked on the paper. As time passed, his expression gradually became serious.

At noon, he came back to his senses in his hunger, but he didn't rush to eat; instead, he picked up his phone and dialed his friend's number. "Hi~ Naris, you're only getting up now too? Were you drained by the girl from last night?"

The friend asked with a smile.

Naris took a deep breath, his voice trembling slightly, "The Landau-Siegel Zero Conjecture has very likely been proven!"

"Naris, this joke isn't funny; you know I'm working on this topic... Where did you see the paper?"

The sound of clothes being put on already appeared on the other end of the phone; he seemed a bit panicked.

"Arxiv."

"Arxiv? Hahaha, don't talk about the Landau-Siegel zero; aren't there several papers about the proof of the Riemann Hypothesis every month?"

On the other side of the phone, the friend seemed to breathe a sigh of relief.

"The author of the paper is Xu Qingzhou."

"."

The other side fell silent, and after a long while, said: "Give me the link."

Well, no need to panic; it's already beyond saving.

University of California, Santa Barbara.

Study.

A middle-aged man with a Xia Nation face stared at the computer screen with a solemn expression. When he saw the fourth real number sequence introduced by Xu Qingzhou, he couldn't help but sigh, "The younger generation is to be feared, the younger generation is to be feared indeed."

This person, of course, is the mathematician Zhang Yitang.

His proof approach had similarities with Xu Qingzhou's; both had thought of inequalities.

But this young man's proof was more concise and yielded more powerful results.

He took a deep breath.

Zhang Yitang: "For number theorists, there are two universes. In the first universe, the Landau-Siegel zero does not exist, but in the second universe, this zero exists. Today, a young man has led us to find the direction."

The outside world was very lively; everyone was discussing and studying Xu Qingzhou's paper.

Even Xu Qingzhou's side became lively.

First, he received calls from several teachers in Kyoto, and then he received invitation emails for manuscripts from mathematical journals such as *Annals of Mathematics* and *New Advances in Mathematics*. The editor of *Inventiones Mathematicae*, to which he often submitted, even called him personally.

Only after learning that Xu Qingzhou simply didn't have time to revise the paper rather than having an opinion about their journal did they feel relieved.

In the following days, Xu Qingzhou basically stayed indoors, and he temporarily set aside those revision suggestions from arxiv.

There were no logical errors; at most, some formulas could be slightly simplified.

His current focus was entirely on advancing the Riemann Hypothesis.

After more than half a month of effort, Xu Qingzhou derived the first "dragon-slaying technique." He used a function called ξ(z); this function introduced zero density estimation, which is giving a non-trivial upper bound for n(σ, t).

In the calculation, one first needs to separate the real and imaginary parts of the Riemann function, thereby finding a way to make ξ(z) a single-variable real-valued function, in order to study the distribution of non-trivial zeros.

In front of the desk, Xu Qingzhou was buried in calculations when he saw Song Yao walk in carrying a cardboard box. "What did you buy?"

"A weight scale."

Song Yao used a utility knife to open the box.

Xu Qingzhou nodded; they used to have one at home, but it was left at Yandongyuan and not brought over. Seeing Song Yao still opening the box, he wasn't in a hurry and lowered his head to continue the calculation from before.

【ξ(1/2+z)=∫0∞g(t)cosh(zt)dt=u(x, y)+iv(x, y)】

Lemma 9.2: Assume pf(n)ns converges absolutely for σ>σa. If f is a multiplicative function, then ∑∞n=1f(n)n^s=s+f(p2)p2s+

After writing down the formula in a few strokes, Xu Qingzhou stopped his pen, only to find that the living room was eerily quiet, and Song Yao was staring at the scale under her feet.

"Are you alright?"

Xu Qingzhou walked over; the school beauty Song looked like she had suffered a great blow.

"I've gained three jin..." Song Yao was listless.

Xu Qingzhou glanced at the weight scale: 98 jin.

He smiled and comforted her: "This is healthy. You're 166 cm and only 98 jin; that's considered light."

"I'm 168 cm."

Song Yao corrected him with wide eyes, paused, and added: "Even without shoes, I'm 168 cm..."

Three years ago, she had indeed exaggerated a little bit.

Xu Qingzhou scanned Song Yao from top to bottom and nodded slightly, "It is bigger than before."

"Huh?"

"You've grown a lot taller."

Song Yao rolled her eyes at Xu Qingzhou, took the vegetables she bought in the afternoon into the kitchen, and a moment later poked her head out again, "Starting today, I'm going to quit milk tea, quit late-night snacks, and absolutely not binge eat. Xu Qingzhou, you supervise me."

She had made up her mind to lose weight.

Two or three jin; there was no need to mistreat herself at breakfast or dinner. Anyway, she would go running every week, so she should be able to lose it.

Xu Qingzhou understood a girl's obsession with weight; controlling her diet was fine, as long as she didn't starve herself.

He said: "That's fine, but there are some bonuses from Jingda and MIT. I originally planned to take you out for a big meal—the Union Oyster House you mentioned last time. The seats are already booked, and now..."

The Union Oyster House is a landmark building in Boston and can be said to be one of the oldest restaurants in the United States that has been in continuous operation since it opened. Since 1826, the building it occupies has had a history of over 300 years.

It's quite interesting.

The two had talked about going to see it before, but never had the time.

Jingda gave a 200,000 yuan bonus, and MIT gave 50,000 US dollars. A huge sum of money had arrived, just in time to go have a good meal.

Song Yao was washing vegetables and became conflicted upon hearing this, but still chose the food: "Then let's eat before dieting."

Xu Qingzhou couldn't help but be amused; she was just swearing to lose weight, but this girl was quite capable of letting go.

Song Yao poked her head out again, looked at Xu Qingzhou on the sofa, and said irritably: "What are you laughing at? Hurry up and come peel the garlic."

"Oh, alright."

December 25th.

Outside the window, heavy snow was falling, but it still couldn't cover the brilliant lights in the distance. Today is Christmas, and the whole city is bustling. The banks of the Charles River are hung with colorful lights, and tourists are like weavers.

In the living room, next to Er Miao's little nest is a small Christmas tree that the school beauty Song bought back.

Xu Qingzhou and Song Yao followed the local customs and celebrated a little bit—roast turkey and smoked ham, the sense of ritual was quite sufficient.

The two even drank half a bottle of red wine incidentally.

Xu Qingzhou came out after washing the dishes, and the school beauty Song was holding a pillow, typing away, chatting with Ding Jiahui.

Guo Ziyang and Ding Jiahui had planned to come to the US for a trip a few months ago, but plans couldn't keep up with changes. Both of them had to go to company internships and write graduation theses, and they were busy as dogs every day, without even a holiday, let alone traveling abroad.

He habitually went over to pull at Song Yao's long legs.

Xu Qingzhou pinched them a few times and couldn't help but suggest: "Tonight, you should put on stockings."

"No."

Song Yao refused without raising her head.

"Today is a special day, suitable for wearing stockings. How about that white pair?"

Song Yao couldn't be bothered to pay attention to this lecherous thing.

Xu Qingzhou still wouldn't give up, "Wear them, wear them; maybe when you wake up tomorrow morning, there will be many gifts in the socks."

Song Yao's typing hand paused, she looked up, and once again revealed an expression of looking at a fool.

Whether there would be any gifts, she didn't know, but she could be certain that after wearing them, there would definitely be some dirty things on them.

"You don't think this reason is good? I can change to another one... Look, you're going on a business trip again in a while; shouldn't you take advantage of being at home, hmm?"

Song Yao rarely had an 8-day holiday, after which she would need to go to Wall Street with Professor Morton. Currently, Professor Morton is employed as a risk strategy analysis expert for the famous financial company Morgan Stanley.

"The business trip is ten days from now."

Song Yao pouted.

Xu Qingzhou's eyes lit up, "You mean, we can do it then?"

"No!"

Xu Qingzhou shrugged, not in a hurry; anyway, there was still New Year's Eve later, and there was still half a bottle of red wine left.

Wine gives courage to the cowardly; let's see how the girl escapes then.

Song Yao chatted with Ding Jiahui for a while, then pulled out a stack of materials from the side. Before reading, she reached out to pull Xu Qingzhou over, gave him a peck, and hooked his chin, "Xu Laoshi, just wait for me to keep you as a kept man in the future."

"Truly impressive."

Xu Qingzhou echoed, praising the school beauty Song's grand ideal.

He stared at the ceiling, pondering the Riemann Hypothesis.

This thing requires a very wide range of mathematical tools, starting from the limit problems of mathematical analysis, to integration, differentiation, series, etc., then to complex variable functions, and then to functional analysis, numerical calculation, integral transforms, and so on.

Fortunately, he has a wide range of knowledge...

Besides his direct proof path, there have been other methods in the previous hundred-plus years.

Because directly proving the Riemann Hypothesis is very difficult, some mathematicians chose another path: outside the line where the real part equals 1/2, the non-trivial zeros of the z-function are very few—so rare that there are at most n of them.

This means that when n here is reduced to 0, the Riemann Hypothesis is proven.

But unfortunately, this is also a path of no return.

During the Christmas holiday, the two lazed at home for two days and then each got busy with their own things—one sank into complex calculations as if possessed, and the other spent every day looking at Morgan Stanley's company materials.

Four days passed. On the banks of the Charles River, two young people were walking slowly. Having been cooped up at home for many days, seeing the weather was nice, they came out to exercise.

"Buy something casual for dinner?"

Xu Qingzhou smelled the fragrance of fried snacks and looked at the mobile snack cart in the distance.

"I want to lose weight." Song Yao's attitude was very firm.

"Really not eating?"

"Not eating."

"Teacher Song, you are very good right now. No, to me, you are perfect; you really don't need to create anxiety for yourself."

Song Yao was unmoved.

"Besides, from the 26th until now, you've already been dieting hard for 4 days; can't you indulge yourself occasionally for one day?" Xu Qingzhou was still talking.

Song Yao raised her eyebrows, glanced at the direction of the snack cart out of the corner of her eye, and was a bit tempted, but thinking that she had already eaten and drunk heavily on Christmas, she quickly withdrew her gaze.

Control the mouth, move the legs.

"Take a step back, you even ran 9 kilometers today..."

"Xu Qingzhou!"

"Fine, I won't say anything."

(End of chapter)