Week 7: The Riemann Hypothesis / by Valzorra

Overall, Week 7 was rather slow as it was Reading Week, which meant that most of the week was free from any workshops or lectures. I took this time to catch up on some of my project proposals, which have already been published, and to document bits of research I did that are quite significant to Project Proposal 4 in particular. I will take this moment to note that the research itself was done last week, however, I am only now getting about to documenting it. Hopefully, it will all make sense and click together once laid out. Additionally, I wanted to take the opportunity to summarise and reflect on what Andy said during our one to one. Now, without further ado, let’s get into the research and updates.

The Riemann Hypothesis

The Euler–Riemann zeta function plays a crucial role in modern analytical number theory and has a variety of applications spanning across fields such as Probability Theory (*wink wink*), Physics, and Statistics. It’s basically a function whose argument can be any complex number other than 1, and whose values are also complex. Euler first studied this function as a real variable and was able to work out its values at even positive integers. In fact, the first even positive value of the function provides a solution to the Basel Problem. Riemann then expanded on Euler’s analysis of the function and established a relationship between its zeros and the distribution of prime numbers. What’s more is that from the Euler-Riemann zeta function stem a variety of other number series such as the Dirichlet Series and L-functions.

The real part and the imaginary part of the Riemann Zeta Function at the critical line.

Now that I have provided a very basic overview of the Euler-Riemann Zeta Function, I can move on to briefly describing the famous Riemann Hypothesis. The Riemann Hypothesis proposes that the Euler-Riemann Zeta Function has all of its zeros at negative even integers, which are all trivial zeros, and complex numbers with real part equal to 1/2, which are the more exciting non-trivial zeros. As the Euler-Riemann Zeta Function is closely connected to the distribution of prime numbers, if this hypothesis is to be proven it would completely revolutionise the way we interpret modern number theory and pure mathematics. What’s important to note here is that if this unsolved problem is proven it could open a series of doors to new ways we can think about mathematics and apply them in the sciences and in invention. Solving this problem would change the way encryption and computer system security functions fundamentally, which may be a reason why some might not want a solution to be found.

The Riemann Hypothesis has a series of parallels to the world and events examined in Project Proposal 4. In that universe, the equivalent of the Riemann Hypothesis, is the problem of humans only being able to use one implant, which is determined genetically. Even though there are a total of seven Alteration Implants developed, no one has managed to crack how one individual can use multiple at the same time. That is the major unsolved mystery of the time and the narrative of the game would revolve around what it would be like if such an important problem were to be solved. Additionally, one reason the protagonist is called Zeta is to echo Riemann Zeta Function and how she essentially managed to solve her universe’s equivalent of that problem. More details on the specific narrative will follow soon, however, even thus far I thought it was a nice nod to this area of mathematics and the potential consequences it may have.

Riemann, 1859

Andy’s Feedback

The Friday of Week 7 was dedicated to one to one sessions with Andy, and I was quite excited to hear what he had to say. As I was most fond of that idea, I presented Project Proposal 4, the game involving dice and managing environments based on what rolls. I gave Andy a very brief overview of Zeta’s world, of her character, and of the main dice-controlling mechanic. I described the special abilities I thought would be appropriate and how one would be able to shift between them. When I was finished with my explanation, Andy seemed pleased with the idea and even mentioned he is not completely sure what to add on to it. He asked me what I was most concerned about at that stage and I mentioned that although I have figured out how the world would work, I have not yet designed an actual level. Andy did not seem incredibly concerned about this and reassured me by saying that it sounds like a game he would like to try. He also recommended a book called The Dice Man by Luke Rhinehart, which revolves around a man who makes most decisions in his life by the roll of a dice. Overall, I was rather pleased with the feedback as Andy was not really able to punch any holes through the idea, which hopefully means that it’s rather solid. I look forward to getting more feedback on it from Adam and my course mates.