Ever since I was introduced to Ulam's Spiral in May, the Riemann Hypothesis, a problem that fascinated me (but which I had decided was too hard to bother with), has had me in its grip. Perhaps it was the confidence I gained after working on the congruent number problem with another professor that made me realize if I really wanted to, undergratuate or not, I can learn what I need to and have some fun. So that's what I've been doing.
My work on the Riemann Hypothesis has come in many growth spurts. Each time, it's felt like I am getting closer and closer. The original exploration came from a unique choice I made in how to represent the sum, that later an attempt to understand it geometrically (in images that could be explained to a classroom) led to another interesting connection. At the beginning of this month, I had a conjecture, based on some of the data I had gathered, but it wasn't solid enough.
The term has been very busy, so both the RH and my other explorations have only come out now and then - usually when I have an idea that I simply cannot ignore. Last night - or should I say this morning - was one of those times.
Prior to this, I had been exploring some of the properties or composite expansions that naturally cropped up in the unique way I had written the zeta function. I noticed that there were certain types of series that could be written to span all the numbers - much like the proof of Euler's infinite product series.
It was 1:30am, and the idea came to me like a dream. For a while I did math in my sleep, then I realized I was awake, and the moment was now. Long story short, by 6am I had an outline for how to turn that idea into a proof, and have been exploring it on and off today, in between study sessions for my courses. I had a chance to run my reasoning by a professor who was familiar with series analysis, and he encouraged me to follow up on it.
This is very exciting! However, I can't be sure that the correct series can be found easily (if at all). I have some rigorous (and careful) expansions to do, some trial and modification, and perhaps some more surprises from intuition, but regardless of where this goes, there's no doubt it's going to be fun.
And, I must add, it's keeping my trigonometry and algebra skills sharp!