Chapter 10
Mom stopped washing the dishes and looked back.
“Paper? What paper?”
“Actually, I already proved the Riemann hypothesis. I want to write a paper about it.”
Clang-!
At that moment, the plate in Mom's hand dropped to the bottom of the sink.
“Really? No, really?”
Wide-eyed-.
Mom's eyes went round.
Nod-nod-.
Mom must also know just how hard and incredible the feat I'd accomplished was.
Grin-.
Seeing that expression, I couldn't help it and a laugh escaped me.
“Heh. I'm amazing, aren't I?”
I puffed my chest out proudly and thumped my chest with my fist.
“Wait a second, Ayeon. Tell me in detail.”
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We sat down at the living-room table, the three of us.
Mom's hands were still a little wet, and Dad looked like he'd been summoned without knowing why.
I sat with both hands neatly placed on the table in front of me.
“Ayeon, can you tell me in detail?”
I explained it step by step.
In my previous life, I solved the Riemann hypothesis right before I died.
The night before I died, I summarized the key idea in the cloud and saved it.
So I wanted to write a paper about the proof.
I said it in a calm, matter-of-fact tone.
The reason I hadn't said it earlier was that I'd been too overcome with emotion, bawling my eyes out, and hadn't been able to explain it properly.
“But my finger muscles aren't developed enough to type by myself... so I need Mom's help.”
Silence fell for a moment.
Mom closed her eyes and said quietly,
“…I see. So in the end, the professor really did it.”
More of a murmur to herself.
Mom knew what the Riemann hypothesis was, after all.
She must have felt that firsthand while reading related papers during her master's program.
How difficult the problem was.
Why every top mathematician in the world had tried and given up.
There were even urban legends that some mathematicians who took on the problem developed schizophrenia and others dementia... a devil of a problem, they said.
While Mom was thinking quietly, Dad spoke up.
“Wait. You solved the Riemann hypothesis?”
“Yep!”
“That's the problem Professor Shin I-jun had been stuck on since his PhD, right? You solved it in the end?”
“Yep, right. My past-life self solved it right before dying.”
“That's our girl! No, wait—wasn't it when you were Yoo Ayeon? Either way, you did great, so come here!”
“Eek! Heehee!”
Whoosh-whoosh-.
Dad grabbed me by the sides and gave me a first-class airplane ride.
As my body lifted into the air, my semicircular canals reacted violently and my brain went into an excited state, which made me giggle uncontrollably.
Once airplane time was over, Mom, having finished thinking, said,
“All right, Ayeon. I'll help you.”
My eyes lit up.
“Really?”
“Yes. Because it's my daughter's request and no one else's.”
Mom said with a gentle expression.
“Yay—!”
I shot both hands into the air and shouted hooray.
“But there is one condition.”
“What's that?”
Mom looked straight at me.
It was a serious expression, different from before.
“You eat on time. You go to bed early. And you never skip time with the family. That includes Mom, Dad, and Arin.”
“…That's a given.”
“It isn't a given.”
Mom's voice turned firm.
“Do you know how Professor Shin I-jun lived while researching the Riemann hypothesis? He stayed in the lab for three months without coming out. No sunlight, no exercise, takeout only, chugging high-caffeine drinks by the case, hardly any sleep, and even then he'd curl up on a hard sofa and sleep like a shrimp.”
“…“
I shut my mouth.
Because I had nothing to say.
“And what happened in the end? He already had an underlying illness, and in the end his health worsened until he died of a heart attack. That wasn't just an unlucky accident.”
I pictured the moments right before my previous life died.
The lab piled high with takeout containers and cans of caffeinated drinks. The days of nonstop wakefulness. And then, on the day the proof succeeded, that sudden pain crushing my chest.
I already knew now that the heart attack hadn't come out of nowhere.
He definitely hadn't been like that when Mom was there as a master's student, but once she graduated and left, he lived in the empty lab like a runaway colt.
“Back then the professor was alone, so nobody could stop him. I won't let the same thing happen again. As long as I'm here, Ayeon will never be allowed to live like that. Got it?”
“…Yeah, I get it...”
I answered quietly.
‘Eek... I got scolded out of nowhere...’
Pat-pat-.
Mom patted my head.
“Our Ayeon answered well. Good girl.”
Dad nodded beside her with his arms crossed.
“Dad agrees too. You can take your time with the paper, but you can't skip meals or sleep. Come on, pinky promise with Dad.”
Dad held out his pinky finger.
“…Ugh-! I get it already!”
At my parents' repeated emphasis, I puffed out my cheeks a little.
‘Seriously, who do they think I'm talking to, a baby?’
Even as I grumbled to myself like that, I didn't forget to hook my pinky with his.
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Since Mom and I started working together, our routine became like this.
First, I organized the paper's draft in my head.
Of course, it was going to an international journal, so the assumption was that it would be written in English.
We worked on the paper during Arin's nap time.
Mom opened her laptop, and I sat right next to her.
My job was to speak, and Mom's was to type.
At first, I thought there'd be no problem, but an unexpected issue made things much harder.
“Uh, a Hilbeot space is defined as—”
“Wait, wait. What?”
“A Hilbeot space.”
“…Hilbeot?”
“Hilbeot! Hilbeot space!”
“Could you mean Hilbert space?”
“Yeah! That one, that one!”
That was because my tongue was too short, so I couldn't pronounce English properly...!
My th sounds were flattened into d's, my r sounds were half dropped, and there was no distinction between l and r at all.
Because of that, it was impossible for Mom to transcribe my English exactly as I said it.
So from Mom's perspective, she had to first figure out what I meant, reconstruct the correct English in her head, check it with me, and only then type it out.
Naturally, it had to be slow.
“Then should I type, ‘A Hilbert space is defined as a complete inner product space.’?”
“Yep! Exactly!”
“But, Ayeon. If I don't know why this space is defined that way, I can't type the next sentence.”
“Why?”
“Because Ayeon's English pronunciation is so cute, I don't think I'll be able to make out the next sentence either.”
Mom said with a little chuckle.
Pssh-.
Realizing my English pronunciation wasn't good, embarrassment washed over me.
My face burned.
“Ugh...”
“It's okay, Ayeon. You're still young, so your tongue is just a little short. Your pronunciation will improve quickly as you grow.”
Pat-pat-.
Mom said, patting my head.
“Sniff, but still...”
Typing exactly what I said was absolutely impossible.
If I wanted to write the paper the way I wanted, I had to make Mom understand me.
‘Mom. A Hilbert space is a complete vector space equipped with an inner product. Think of it as the vessel that will hold the operator whose eigenvalues are the zeros of the Riemann zeta function. Without that vessel, there's nowhere to define the operator.’
I had clearly intended to say exactly that.
But my pronunciation slipped, so it didn't come across correctly.
“Hmm... I heard 'queen,' but did you say complete vector space?”
“…Yeah.”
“And what did you say after 'the zeros of the Riemann zeta function'?”
“Eigen... eigenva...”
“Ah. Eigenvalues? I get the gist of it. I understand the rough idea.”
“If you get it, then type the next sentence.”
“Okay, okay.”
Clack-clack-.
That was how it went.
For the paper, I first had to make Mom understand the logic.
But Mom only had a master's and hadn't even gotten a doctorate, so she couldn't understand the sentences in concrete detail.
And since my pronunciation wasn't accurate either, getting it across to Mom was pretty hard.
‘I was prepared for a long battle...’
But with such a severe bottleneck, it was frustrating.
Haa-.
I let out a short sigh.
But what else could I do?
I couldn't show Mom my brain.
Even if it was inconvenient, I had to keep moving forward slowly.
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I'll explain some difficult stuff for a moment, so readers whose heads hurt while reading can just skim past it. Okay?
The Riemann hypothesis is a problem about the hidden pattern of primes, and while it seems like a number theory problem that deals with primes a lot...
the function can be transformed in many different ways, so there are various approaches.
The direction I took was a physics-based one.
It's called the Hilbert-Pólya conjecture, an approach that turns it into a quantum-mechanical energy-level problem.
This is also why many scholars see the Riemann hypothesis as essentially a physics problem.
The structure of the paper was like this.
First, I had to define the proper Hilbert space.
A space that could handle the zeros of the Riemann zeta function.
H = L^2(X, dμ)
That choice itself formed the foundation of the entire proof.
Then I constructed an operator H that satisfied specific conditions.
(Hf)(x) = i * (d/dx) f(x) + V(x) f(x)
This was the most important part.
The reason the Riemann hypothesis had remained unsolved for the past 160 years was that nobody had found this.
This operator's eigenvalues all had to be real, and they had to match the zeros of the Riemann zeta function exactly.
My past-life self spent three months pouring everything into this.
When I explained this part, Mom shared what she thought.
“This approach feels kind of physics-like... though I don't really know physics.”
“Still, this is the most important part, Mom.”
“The part where you define operator H?”
“Yep. If I can show this H satisfies the three conditions, the proof is over.”
“Really? It's that simple?”
“Yep. Of course, each step required a massive amount of writing.”
I spread my arms wide to emphasize how huge that was.
Of course, operator H couldn't just be anything; it had to satisfy three tricky conditions.
Self-adjointness.
I had to show that H's adjoint H* was exactly equal to H.
Spectrum.
I had to show that every eigenvalue of operator H matched the function's zeros perfectly.
Trace Formula.
I had to prove that the spectral (zeros) expression and the geometric (prime) expression were identical.
So, since finding an H that satisfied those three conditions was so difficult, even with a $1 million bounty on it, nobody had solved it until then.
“Mom, do you get it?”
“…No. I don't understand a single word of it.”
Pout-.
She looked downcast, uncharacteristically so for Mom.
Well, of course.
When Mom was my student, I'd go “good, good” and pat her head a lot.
I only had her study things suited to her level and gave hints for the difficult parts.
By contrast, what we were dealing with now was content so difficult even other professors would struggle to understand, let alone someone without a doctorate.
No matter how gifted a genius Mom was, it was impossible for her to understand this right away.
Since I was landing real UFC blows instead of staged WWE moves, it was only natural she'd feel dazed.
“To think I can't even understand what my own daughter is saying... what a shame for a mother...”
“It's okay, Mom! Your job is just to copy down the English sentences I say! You don't need to understand everything!”
Pat-pat-.
I comforted the downcast Mom by patting her head.
Since there was a height difference, I had to stretch my arm way up to do it, which was oddly tiring.
But because she was being comforted by me, Mom seemed even more embarrassed.
“…Ayeon, wouldn't it be faster for you to build up your muscles than for me to understand this?”
“Hmm... maybe? Still, let's type the easy parts first!”
Which would be faster, really? Mom understanding this material, or my pronunciation improving or my finger muscles developing?
Either way, it didn't look like it would be over anytime soon.
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Time flew by like an arrow.
No, come to think of it, an arrow only goes about 250 km/h at best, right?
Yeah. Let's use a railgun with a top speed of 6,000 km/h instead. It looks 24 times faster than an arrow. And railguns are cooler anyway.
Anyway, time raced by like a railgun.
Before I knew it, I was five years old.