The Evolution of Genius: Every Night, I Get Smarter!-Chapter 52: The Revelation

As I strolled into the room, a staff member handed me a microphone, and I nonchalantly pinned it to my blouse.

I glanced at the crowd, full of expectant faces, and cleared my throat. "I’m Maximillian Sullivan, as some of you may already know..." I said with a casual tone, which brought smiles to the audience.

"First, I would like to thank the staff of Harvard for this warm invitation," I continued, prompting the room to erupt in applause.

"Now, I would like to dive into something that has intrigued mathematicians... intrigued US for centuries. The pattern of prime numbers." The audience’s curiosity was picked up.

"Some of you might think that the famous Goldbach Conjecture was my goal from the very beginning, and that’s what I’m here to talk about. But you would be mistaken."

I approached the blackboard.The audience’s collective curiosity hung in the air.

I picked up a piece of white chalk, the dusty residue fell on the floor, and I began to write.

I wrote the formulas and functions of the Riemann Conjecture, which included the Riemann Zeta Function ζ(s)=1^(−s) + 2^(−s) + 3^(−s) + 4^(−s) + ...

As well as the Euler product formula ζ(s) = (1^(−s))(2^(−s))(3^(−s))(5^(−s))(7^(−s))(11^(−s))...

The expansion which connects it to the prime number pattern.

Then, I turned to the audience, "Most of you probably recognize this"

Just as I was about to explain further, a professor with white hair in the front row raised his hand and asked, "Are you trying to tackle the Riemann Conjecture?"

I knew who it was. He was revered as one of the most sought-after Professors at Harvard.

I read his extensive papers on condensed matter theory and string theory.

He held a distinguished reputation as a remarkable mathematician and theoretical physicist.

Professor Dan Freed.

I glanced over at Professor Dan Freed and after a quick thought, I decided it was time to come clean.

I leaned on the large desk before me and said, "No..."

The audience appeared a bit bewildered, some wearing amused expressions, not quite sure what to make of my response.

Professor Dan Freed broke the silence by asking, "Then what’s the point?"

As I faced the intrigued yet perplexed crowd, more questions began to flood in. "Is there a connection between this and the Goldbach Conjecture?" one professor inquired, seeking clarity on my unconventional topic.

"Why aren’t you discussing the Goldbach Conjecture?" chimed in another voice, expressing curiosity about my choice.

The commotion grew, and people in the audience raised various questions. "Tell us about your background. How did you arrive at a solution for the Goldbach Conjecture?" a reporter questioned, holding up a camera.

This triggered a domino effect as more reporters and audience members sought answers.

The auditorium buzzed with an animated discussion as hands reached out, and a loud atmosphere enveloped the room.

With the audience’s questions reaching a crescendo, I raised my hands to quell the commotion. "Hold on, everyone," I urged, "I’ll address your questions shortly."

"Let me clarify something now. No, I’m not attempting to tackle the Riemann Conjecture. In fact... I have already proved it!" The room fell silent.

The audience was in a state of bewilderment, and the professors began firing questions:

When did I prove it?

How?

Did I have the proof in my head?

The disbelief was clearly visible.

I just stated that I have solved the millenium question!

I calmly responded, "I’ve already posted the proof on arXiv on my way here. I wanted today’s topic to be the presentation of that proof."

The room buzzed with a mixture of astonishment and curiosity as the audience tried to come to terms with what I have just said.

Several members of the press in the audience couldn’t contain their excitement and hastily left the auditorium, eager to check for any breaking news or get the first scoop on this astonishing development.

The room was now abuzz with both mathematicians and reporters, all eager to see the proof that I claimed to have posted.

I started explaining the concept to the audience as I approached the three large blackboards at the front of the auditorium.

With a piece of chalk in hand, I began to write down the equations and mathematical symbols, explaining each step as I went along.

The room was silent, and all eyes were fixed on the blackboards.

"Let us consider the Riemann Zeta Function,"

I started, scribbling down the series representation of ζ(s). "Defined as the sum from n equals one to infinity of one over n to the power of s, for real parts greater than one..."

ζ(s)=∑​1/(n^s) ​for ℜ(s)>1

As I continued writing, the professors in the hall followed attentively, their eyes darting between the blackboards and their notes.

"We shall introduce an analytical function F(s), defined on the complex plane, which can be defined as:"

F(s)=π^(s/2)⋅Γ(1−s)⋅ζ(s)

I explained, "Γ(1−s) here, relates the Riemann Zeta Function to the Euler Gamma Function and I can now introduce the Dirichlet Eta Function:"

η(s)=(1−2^(1−s))ζ(s)

"We know that The Euler product formula for the Riemann Zeta Function holds for all complex s with real part greater than 1:"

ζ(s)=p prime∏​{1/(1−p^(−s))}

"To establish the connection with the Goldbach Conjecture," I explained as my hand moved across the blackboard, "we exploit the fact that the Goldbach partitions represent a distribution of even numbers into two prime numbers."

The professors were following along, capturing every step of the intricate proof.

They knew the significance of what was unfolding before their eyes.

"As we examine the distribution of primes," I continued, "we consider the Chebyshev function ψ(x), representing the number of primes less than or equal to x." The room was silent except for the deliberate scratch of chalk on the blackboard.

I took a moment to glance towards the staff and asked, "Hey, could I get a cup of coffee real quick?"

The man was momentarily stupefied, unsure of how to react.

He glanced toward one of the Professors who was overseeing the meeting.

The Professor nodded and said, "Bring the man a cup of coffee."

In a hurry, he quickly left the hall to fetch a cup of coffee for me.

I advanced to the next section of the proof. "By investigating the properties of the Euler product for η(s), we find that it can be expressed as..."

η(s)=1/(2^(s−1))∏​_pprime(1/(1−p^(−s)))=π^(s/2)/sqrt(2)​⋅Γ(1−s)⋅ζ(s)

The audience watched with bated breath as I unveiled the connections between the zeta function, the Chebyshev function, and the Dirichlet Eta Function.

I could sense their collective anticipation, their realization that they were witnessing something extraordinary.

"Now, taking the derivative of both sides of this equation and substituting the Γ(1-s) term, we obtain...",

η′(s)=(π^(s/2))/(sqrt(2))​⋅d​(ζ(s))/ds​

I explained as I carefully wrote down the mathematical expressions.

The professors in the hall were mirroring my calculations, their notepads filled with equations, symbols, and notations.

The room was a sea of deep intellectual engagement.

"By recognizing that the derivative of the Euler product formula for η(s) yields the derivative of the Euler product formula for ζ(s)," I continued, "we have..."

η′(s)=(π^(s/2​)/sqrt(2))⋅ζ′(s)

The professors leaned in, trying to grasp the implications of the connections I was revealing.

"With these expressions, we can equate the derivative of η(s) to the derivative of ζ(s) and then rearrange the terms..."

ζ′(s)=(sqrt(2)/π^(s/2))⋅η′(s)

As the man from the staff returned with a cup of coffee, I paused for a moment to take a slow sip.

I then reached back to the blackboard and continued with the complex proof. "Now, let us focus on the right-hand side of the equation," I said as I moved on to the next step.

"By incorporating the Chebyshev function ψ(x) into the Dirichlet Eta Function, we can write..."

η′(s)=∑∞​((−1)^n)/(n^s)​=−(1−2^(1−s))ζ′(s)

"Finally, combining all these findings, we have... Which simplifies to..."

1−2^(1−s)=−sqrt(2)/π^(s/2)

The professors continued to capture each piece of the proof on paper, their expressions a mixture of fascination and understanding.

"The equation is now suitable for exploring the Riemann Zeta Function outside the region where the real part of ’s’ exceeds 1."

"I’ll denote non-trivial zeros as ’ s = σ + it ’ with σ as the real part and ’t’ as the imaginary part. Now, focus on this. If any non-trivial zeros have a real part different from 1/2... the equation cannot be valid."

"By considering ’ s = 1/2 + it ’ as a non-trivial zero, the equation transforms into:"

1 - 2^(it) = -sqrt(2)/(π^((1/4) + it))

I pointed to the right hand of the equation.

"Notice that the right-hand side possesses a fixed magnitude, while the left-hand side oscillates with changes in ’t.’ Consequently, it is impossible for non-trivial zeros to have a real part other than 1/2"

"Hence, the Riemann Hypothesis is not a Hypothesis. It is a Lemma." I proclaimed.

The hall was quiet.

Some professors were still scribbling notes.

It was so quite that I could hear my own heartbeat.

I turned around and faced the audience confidently, "This concludes the proof"

Everyone in the audience was a witness.

Amazed.

Astounded.

Incredible.

Everyone in the audience at Harvard was left in awe.

A profound silence filled the lecture hall.

An elderly professor was the first to break the hush.

He rose from his seat and started to applaud.

Clap clap clap...

The staff member that stood next to the podium could not understand the proof process on the blackboard, but he could not help but clap as well.

As I reached for my cup of coffee and took a sip, numerous cameras took pictures, with me on the forefront.

’Harvard Lecture Hall Erupts in Applause as Riemann Hypothesis is Proven’

’Maximillian Sullivan, Possibly The Greatest Genius Of The Century’

’18-year-old Mathematician Solves The Millenium Question’

’The Next Fields Medal Guaranteed To Go To The Young Genius?’

These were the headlines that captivated the world for the following month.