One Hundred Meetings
Today, our computation book discussion group is going to have the 100th meeting! Yes, the 100th meeting! We began these book discussion meetings about five months ago. The first book we picked up for our discussions was Introduction to Analytic Number Theory by Apostol (1976). We have been reading this book together for the last five months. We have a tiny but consistent community of 6 to 8 participants who meet regularly to study this book and share our understanding and insights with each other.
In this blog post, I will talk about my personal experience hosting these meetings and my personal journey about reading this book. It is worth keeping in mind then that what I am about to write below may not have any resemblance with the experience of other participants of these meetings.
Contents
The Reading Experience
As far as I know, everyone who joins our meetings are involved in computer programming in one form or another. A few of them have very strong background in mathematics. I host these meetings everyday and discuss a few sections of the book in detail. I show how to work through the proofs, explain some of the steps, etc. Sometimes I get stuck in some step that I find too unobvious. Sometimes the steps are obvious but my brain is too slow to understand why the steps work. But these tiny glitches have not been a problem so far, thanks to all the members who join these meetings on a daily basis and contribute their explanations of the proofs.
I believe the group members are the best part of these discussions. Thanks to the insights and explanation of the reading material shared by all these members, I am fairly confident that we are able to take a close look at every proof and convince ourselves that every step of the proofs work.
The Learning Experience
The first web meeting to discuss the chosen analytic number theory book occurred on 5 Mar 2021. See the blog post Reading Classic Computation Books to read about the early days of our group and how it was formed. Back then, I knew little to nothing about analytic number theory. Although I was familiar with some of the elementary concepts like divisibility, Euler's totient function, modular arithmetic, calculus, and related theorems, chapter 2 of the book itself proved to be a significant challenge for me. In the second chapter, it became clear to me that we will be building new levels of mathematical abstractions, use these abstractions to build yet another layer of abstractions, and so on. The chapter began with a description of the Möbius function, a very neat and interesting function that I was previously unaware of. That was fun! But soon, this chapter began adding new layers of abstractions such as Dirichlet product, Dirichlet inverse, generalised convolution, etc. I could almost feel my brain stretching and growing as we went through each page of this chapter.
I often saw that after I have learnt a new concept in a chapter, it would not become intuitive immediately. I would understand the concepts, understand the related theorems, understand each step of the proofs, solve exercise problems, know how to apply the theorems when needed, and yet I could not "feel" them. I wanted to not just understand the concepts but I also wanted to "feel" the concepts like the way I could feel algebra, calculus, computer programming, etc. In the initial days, I wondered if I was too old to develop good intuition for all these new and highly sophisticated concepts.
Despite always feeling that all these concepts were too technical and quite unintuitive, I kept going. I kept hosting these discussions with a frequency of about 3-5 days every week. We continued discussing the various chapters and the proofs in them. And then suddenly one day while reading chapter 4, something interesting happened. As we were employing Dirichlet products to obtain some useful results, I realised that the concept of Dirichlet products which once felt so foreign two chapters earlier, now felt completely intuitive. I could see different functions being equivalent to Dirichlet products intuitively and effortlessly. Dirichlet products felt no more alien than, say, arithmetic multiplication. I could "feel" it now. It was a great feeling. I realised that sometimes it might take a few additional chapters of reading and using those concepts over and over again before they really begin to feel intuitive.
Three Concepts
In this section, I will pick three interesting concepts from different parts of the book to provide a glimpse of what the journey has been like. These three things occur in the book again and again and play a very important role in several chapters of the book. Of course, it goes without saying that there are many interesting concepts in the book and many of them may be more important than the ones I am about to show below.
The Möbius Function
For any positive integer
I was unfamiliar with this function prior to reading the book. It felt like a nice little cute function initially but as we went through more chapters, it soon became clear that this function plays a major role in analytic number theory.
As a simple example, we will soon see in this post that the Euler's
totient function can be expressed as a Dirichlet product of the
Möbius function and the arithmetical function
As a more sophisticated example, the Dirichlet series with
coefficients as the Möbius function is the multiplicative
inverse of the Riemann zeta function, i.e., if
Dirichlet Product
If
Hurwitz Zeta Function
For complex numbers
Now admittedly, the definition or the analytic continuation of Hurwitz zeta function may seem very heavy and obscure to the uninitiated and it is indeed quite heavy. It takes 6 pages in chapter 12 to build the prerequisite concepts before we arrive at this definition. It is evident that this definition uses other concepts like the gamma function, a specific contour integral, etc. and it is only natural to expect that one has to gain sufficient expertise with the gamma function and contour integrals before the Hurwitz zeta function begins to feel intuitive.
But once we have established the analytic continuation of the
Hurwitz zeta function, many insightful facts about the Riemann zeta
function follow readily. It is easy to see that the Riemann zeta
function can be defined in terms of the Hurwitz zeta function as
I believe beautiful results like these obtained by digging deep into complex analysis are what makes the study of analytic number theory so rewarding.
The Next Meeting
The next meeting is coming up today in a few hours. Are we planning anything special for the 100th meeting?
I think the 100th meeting is a significant milestone in our journey of understanding the beautiful and interesting gems hidden away in the subject of analytic number theory. This milestone has been possible only due to the sustained curiousity and eagerness among the members of the group to learn a significant area of mathematics and learn it well. We have reached this milestone successfully due to the passion and love for mathematics that drive the regular members to join these meetings and go through a few pages of the book everyday. In these meetings, we have read 12 chapters consisting of over 250 pages so far. Many of us knew nothing about analytic number theory merely five months ago and now we can appreciate the Riemann zeta function at a deeper level. We now understand what the Riemann hypothesis really means. This has been a great journey so far.
Despite being a significant milestone and cause for celebration, we
are going to keep our 100th meeting fairly simple. We will continue
where we left off yesterday. Today we have some more relationships
between the gamma function and the Riemann zeta function to go
through, so that is what we will do. We will also show that
Join Us
If this blog post was fun for you and you would like to join our meetups, please go through this page to get the meeting link and join us.