Seven Proofs of Riemann’s Hypothesis: A Short Story

In Mathematics, Riemann’s hypothesis (RH) is one of the unsolved problems. It is one of the Millennium Prize Problems. Whoever solved this problem would be awarded one million dollars by the Clay Mathematics Institute. It states that all the non-trivial zeros of the ‘zeta function’ lie where the real part of the argument of the zeta function has value one-half. This line is called the critical line. Normally, an unsolved problem in mathematics is called a conjecture. But this one is called a hypothesis because the proofs of many theorems depend on it. If RH, then … .

I learned about RH when I was a graduate student at Quaid-i-Azam University (QAU), Islamabad. One day I was sitting at the Chemistry Huts – a tea shop – and thinking of RH. Chemistry Huts is one place where we girls hung out during breaks between classes. But today I was alone. I ordered chai with samosas and sitting on a bench on the far side of the huts away from boys. In Pakistan, if you are a girl and on top of it have white skin, your life is like hell. I also had many stalkers who followed me to my doorstep. People did not give me a break asking my hand from my parents. But my answer was a big NO. I didn’t want to marry yet. I know if I proved RH, I would have many offers by foreign universities. But I want to go to Princeton University, my favorite university.

Although I ordered tea, I lost appetite and feeling nauseous. I got up and canceled the order and went straight to the Department of Mathematics looking for Prof. Mukhtar Hussain. Though I’m majoring in physics, I took several courses in the mathematics department. Dr. Hussain taught us a course on number theory. This was where I learned about Riemann’s hypothesis.

It was about 3:00 in the afternoon. The department was almost empty. Most students were at the ‘point’. The point was a place where the blue university buses leave. Whether one was leaving or not, he would reach to point for ‘phondi’ – checking out. Popular times were 1p.m., 3 p.m. and 5 p.m. I left at 5 p.m.

I went to Dr. Hussain office and knocked on the door.

“Yes, please come in.” The professor said from the inside.

“May I come in,” I said.

“Hi Hafsa, how can I help you?”

“I have proved the Riemann’s hypothesis.” I abruptly said.

The professor scanned me from toes to head and offered me to prove it on the whiteboard in his office.

I proceeded to the whiteboard and drew a horizontal line – which represented the real part of the argument of the zeta function – and two vertical lines. The first vertical line represented the imaginary part of the argument of the zeta function and the second vertical line represented the critical line. I drew a little circle on the critical line and said, “If one non-trivial zero lies on the critical line, then all the non-trivial zeros must lie on the same line.” Before that the professor commented, I continued, “But I’m not going to publish my paper until I have seven proofs because the title of my paper is, “Seven Proofs of the Riemann’s Hypothesis.” So far, I have one proof. My paper would be at least 150 pages, I can tell. The tougher is the theorem, the longer is the proof.”

The professor was sometimes looking at me and sometimes at the diagram that I drew on the whiteboard. “Hafsa, are you okay? Your eyes are red. It Looks like you haven’t slept for days. You need to rest.”

“I’m OK, Sir.”

I knew the professor didn’t take me seriously. This professor was also a candidate. He also proposed to me. Where to go? Who should I discuss my work with? More than my work, people are interested in me. I’m sick of it. I don’t want to be known by my beauty but by my talent.

Of course, I didn’t have seven proofs yet, but I clearly saw it: analytic proof, elementary proof, topological proof, geometric proof, binary proof, categorical proof, and poetic proof.

The next day, the news spread in the university like fire. People were ridiculing me. I was called ‘Seven Proofer’. I was not sure whether Prof. Mukhtar Hussain spread the news or how, but one thing was clear that people kept news about me instant by instant. People’s interest was to know who I liked. Now Prof. Hussain was in the news that I liked him.

The idea of seven proofs came in mind at the end of September 2001. Events were changing very quickly. One day, in early October, I was in the class that all of a sudden, I left the class, and with chalk, I started drawing a diagram – what is called the commutative diagram in category theory – on the floor in the hallway. As soon as I completed the diagram, I got up and shouted pointing at the diagram, “This is the categorical proof of RH.” During this, students left the class and they were observing me. Not only students but the crowd around me also included professors. To my surprise, Prof. Hussain was also present. Other people were quiet, while Prof. Hussain was putting his thumb under his chin and smiling. I understood he was impressed. But I did not know who informed him that I was here. He was a professor in the mathematics department whereas this was the physics department. This place where I was standing was the second floor. The department was like a square. Each corner is called a wing where there were offices for professors and research students. The ground floor could be seen from here. In front of me on the opposite side was the Chair office. The Chair whose name was Prof. Amir K. Dar was also watching me. People on the ground floor were also observing. Then suddenly the people, whether upstairs or downstairs, started to blur and disappeared in a wavy manner. I did not see anything but a white sky. I was frightened. Then gradually people started to appear again. But this was a different place. Someone told me that that was Princeton University.

“Your seven-proof-paper has published. You are recognized. Princeton University has offered you a Ph.D. position in mathematics.” Prof. Hussain said to me.

“What! Really! But what are you doing here? Aren’t you a professor in QAU?”

“I’m also a Visiting Professor at Princeton.”

Prof. Hussain started to disappear gradually in a wavy manner. I was again seeing a white sky. The white sky then started to fill with people. This was again the physics department in QAU. Perhaps I was daydreaming.

Over time the craziness faded away. I realized that Riemann’s Hypothesis might never be proved at least in my lifetime. The lesson I learned from the madness was to switch from physics to mathematics. I published several papers in pure mathematics.

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