“S-Seoha?”
Theo looked bewildered.
When the proof ran into an obstacle, Seoha’s atmosphere changed. His gentle demeanor was nowhere to be found; there was even a hint of madness in his face.
Apparently he was not the only one who sensed the change, for a low murmur rose in the lecture hall. But Seoha heard nothing.
Wuuung—
The world only hummed like a gigantic beehive.
Seoha took a step back and looked at the entire blackboard. Equations filled three sides of it. They entered his eyes like a vast landscape.
‘I can’t see it from here. Let’s go deeper.’
Where did he end, and where did Ducky begin? It was a meaningless distinction. Two consciousnesses were trying to merge into one. Seoha did not resist. At last, the boundary between them vanished.
A new field of vision opened.
There were no straight lines or horizontals. It was a space of noncommutative geometry where everything was curved. Seoha’s body swam within the space created by his own equations.
There, he saw glittering zeroes.
“The stars remain.”
If the operator had diverged into imaginary numbers, they should have disappeared. But they still remained, only trembling in their places.
Suddenly, Seoha recalled reading the waves of the Riemann zeta function in the music room back when he was at the science high school for the gifted. This was it. The intrinsic property of this space.
Real part 1/2.
A single line cut through the space. And all the stars were on that line. They were trembling, but they could never leave the predetermined line.
‘Look.’
Ducky pointed to it.
The gaps between zeroes.
There was something within those intervals that seemed irregular. Seoha magnified his vision and looked at them.
2, 3, 4, 5, 11, 13…
They were prime numbers.
Each time the zeroes aligned, prime numbers asserted their presence between them.
“Are the zeroes drawing in the primes?”
‘Or perhaps the primes are pushing the zeroes away, creating the gaps.’
Seoha expanded the space.
The line of real part 1/2 stretched out infinitely. The zeroes glittered like stars in the Milky Way. He could feel the primes breathing between them.
“Beautiful.”
‘Because it is the rule of the world. We are not the first to come here.’
Ducky spoke quietly.
“Riemann must have seen this too.”
Riemann must have stood alone in this space. But he had no language with which to express it. Noncommutative geometry had not been born, and étale cohomology did not exist. What he held in his hands was only complex analysis and his own intuition.
And so he wrote this in his paper.
[It is very likely that all these zeroes lie on the real part 1/2.]
Darkness slowly descended upon the space.
What had been faint as mist gradually took shape, then formed the figure of a person.
A man of lean build, not tall, with shoulders slightly hunched.
His black coat was neat, but looked somehow worn. A pale face, deep-set eyes. He looked at Seoha with an expression steeped in regret.
Seoha recognized him at once.
“Riemann.”
He took a step closer and said something.
Seoha could not hear his words, but he understood their meaning.
“Were you waiting for me?”
A faint smile appeared at the corners of his mouth. He was relieved.
A peace found after a long wait.
How lonely must he have been?
Seoha felt a tight ache in his chest.
This man had arrived too early. When the world was not yet ready, Riemann had groped toward the boundary of the universe with intuition alone.
He extended a gaunt hand.
Seoha seemed to see ink stains on it. A hand that must have lifted a pen countless times to write equations.
Seoha tilted his head and took his hand.
“Are you inviting me?”
Riemann smiled and shook his head.
Then, as if telling him to listen carefully, he shaped the words clearly with his mouth.
Seoha’s eyes widened.
“Inherit…”
With Riemann’s final words, Seoha’s consciousness was flung back up into the real auditorium.
The cool air of the lecture hall, the thick smell of chalk stinging his nose, and hundreds of gazes fixed on him. Seoha staggered and braced himself against the blackboard.
‘Inherit it?’
The theory he had been forced to lay down his pen before fully writing. A cry telling him to inherit that noble will, which had yearned for the order of the universe even in the face of death.
“Seoha!”
“Seoha!”
His teammates tried to rush forward, but Seoha raised a hand and stopped them. He had to quickly organize what was rummaging through his mind.
Scratch, scratch.
Seoha’s chalk began to move again.
Hundreds of structures appeared and disappeared in his mind. Unsorted intuitions whirled like a storm and collided with one another.
‘Don’t worry, Riemann. I can do it. Mathematics has advanced a great deal since then.’
Among those many intuitions, Seoha was ultimately able to find the right path. The space Alain Connes and Grothendieck had dreamed of. But that alone was not enough. He needed a new geometry.
Suddenly, he remembered the arrangement of the zeroes he had seen in Riemann’s space and the primes within them.
From 2 to 3, from 3 to 5… They were not a mere sequence of numbers. They were an arithmetic flow in which the wave of the preceding prime determined the phase of the next prime.
‘Therefore, the operator…’
Seoha took over Ducky’s thought.
“It was a metronome controlling waves and the flow of time.”
A geometry that could contain this.
The mistake had been trying to fix numbers in space. A static space could not contain time and waves. What appeared in Seoha’s eyes was a sea of numbers where enormous energy surged.
Scratch, scratch.
Realization surged in.
In the world of waves, order itself becomes truth. Because an irreversible arithmetic time flows within the current of primes.
In this bizarre space Seoha had created, even a point became part of the flow. In other words, the operator was inevitably!
“Huh?”
Andrew was the first to notice the anomaly.
A geometric space he had never seen before. A higher-dimensional concept encompassing spacetime. Naturally, it was a place where absurdly immense chaos existed.
When Seoha’s operator dropped anchor upon the rough waves there, the imaginary terms that had been diverging and denying logic were miraculously canceled out and vanished.
Whitman stroked his chin as if pondering something.
“It resonates?”
Andrew nodded.
“I think that space will require further verification, but if Seoha’s logic is correct, that operator is the Hamiltonian of this geometric universe.”
The professors could not come to their senses.
Seoha had been deep in thought, then suddenly began writing something down with ferocious speed. From that moment, most of the professors could no longer follow the flow.
And now, the Hamiltonian.
The fantastical key that had been Hilbert and Pólya’s longing. This operator possessed the property of being self-adjoint.
It meant that energy was perfectly conserved in the process by which the wave called a prime number passed on to the next prime. If so, naturally, every zero of the zeta function would be unable to leave a single line.
A proof in the most beautiful form, one that allowed no rebuttal.
Gulp.
The atmosphere of the lecture hall changed strangely.
Those who understood and those who did not. Some stared even more intently at the blackboard, while the majority of the scholars looked around, unable to grasp what was happening.
“It seems we’ll be heading toward the end soon.”
“To think I would see this before I die. Truly, it pays to live long. This is the greatest fortune of my life.”
“I should have taken my medicine first. I’m worried my heart may stop.”
The final moment of verification remained.
If the zeroes of the zeta function coincided with the spectrum of this operator, the proof would be complete.
Scratch, scratch.
Seoha wrote a single equation on the blackboard.
The condition under which the zeta function became zero. Seoha translated the Riemann zeta function into the language of the geometric space he had created.
The terms of the function changed one by one.
“Th-That!”
“Is this transformation really correct?”
“Please be quiet.”
Andrew stared blankly at the blackboard.
The transformed terms matched the spectrum exactly. Seoha’s logic was perfect. Because the structure did not allow it, nothing could escape from here.
Tap.
The final equals sign was written on the blackboard.
At that moment, Seoha saw it.
A carpet was laid across a universe like infinity. A line of the real part stretched endlessly onward. The zeroes embedded upon it glittered like stars in the Milky Way. And within it, an operator pulsing along the primes.
It was a breathtakingly majestic and beautiful sight.
Seoha closed his eyes.
The faint apparition of Riemann appeared. He turned up the collar of his old coat, then nodded as if satisfied. Then he slowly disappeared into the deep darkness.
The masters who understood Seoha’s proof all wore the same expression.
All of them had reddened eyes and were holding back tears. It had truly been a long, long time. Countless mathematicians had devoted their entire lives to writing this one line of proof.
「…Therefore, all nontrivial zeroes of the zeta function exist on the real part 1/2.」
Hilbert, Pólya, Hardy, Selberg, Deligne, Bombieri, Grothendieck, Alain Connes. And countless geniuses who had vanished without even leaving their names behind.
「Q.E.D.」
Seoha felt solemn.
If even one of them had not existed, could today’s proof have existed? Ask him a hundred times, a thousand times, and he would answer no.
And so this was a tribute dedicated to them. Seoha lifted the chalk and wrote one more line beneath it.
「For Riemann and Hilbert, and for all the giants who live and breathe between the lines of this proof.」
Khhhk.
The moment the chalk stopped, someone’s sobbing could be heard in the lecture hall.
It was Andrew.
White hair, thick glasses, and the hunched back peculiar to a man who had spent his entire life doing mathematics. The elderly professor, respected by the entire world, was shedding tears. Whitman took out a handkerchief and covered his face. The other masters beside him did the same. Their lips trembled, then their shoulders shook as they wept.
Sob, sob.
The sound of weeping spread.
Before long, the lecture hall filled with hundreds of mathematicians had become a sea of tears. No one wailed loudly. Mathematicians did not cry that way.
Out of sorrow and gratitude for those who had departed first without seeing the proof; out of joy that the years of suffering had finally been rewarded; and with respect and thanks for the young mathematician who had inherited the legacy they left behind in full and made it the cornerstone of a new era.
Theo closed his eyes and looked upward. He could not control the tears bursting forth. Amid it all, one thought came to mind.
‘I’m truly glad I was able to share this moment.’
Clap, clap, clap!
The sound of applause came from somewhere.
An old scholar rose without even wiping his wet glasses and began to clap. Then Whitman, and then the winners of the three great mathematics prizes seated in the front row, rose from their seats one by one.
Clap, clap, clap!
The sound of applause spread backward like a wave.
At last, when all the hundreds of people in the auditorium had risen, the sound became as fierce as a storm, as if it would pierce through the ceiling of the lecture hall.
To Sujeong’s ears, it sounded like the trumpets of a military band.
It was praise directed at a genius named Seoha, and at the same time, a victory report that humanity had finally deciphered God’s code.
Reporter Nam felt confused, not knowing how he should describe this situation. He did not know the details, but his heart was boiling.
Yet there were a few things even he could understand.
That something very old had come to an end, that there had been so many people who had waited for this, and that the young mathematician standing before the blackboard had, in the end, answered their waiting.