Our 100th Meeting

Published on 20 Aug 2021 by Susam Pal

Today, we are going to have our 100th meeting! Yes, the 100th meeting! We began these book club meetings about five months ago. The first book we picked up for the club was Introduction to Analytic Number Theory (Apostol, 1976). We have been going through this book, page by page, in each meeting. 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 in this book club 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 members of the book club. In fact, some members in the group are much smarter than I am, so I am pretty sure the experience of other members must be quite dissimilar to mine.

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 the book club meetings everyday and read a couple of pages of the book aloud, 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 the extremely smart group of individuals who join these meetings on a daily basis and help explain why every step of the various proofs work.

I believe the group members are the best part of these reading sessions. 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 line of the book and convince ourselves that every step of the proofs work.

The Learning Experience

The first web meeting for the book club occurred on 5 Mar 2021. See the blog post Hello, World! to read about the early days of the book club 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 on top of new levels of abstractions. 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, generalized 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. We kept hosting the book club meetings with a frequency of about 3-5 days every week. We read one section after another. We read one chapter after another. And then suddenly one day while reading chapter 4, something interesting happened. As we were employing Dirichlet products to obtain some useful results, I realized 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 realized 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 a 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 $$n$$, the Möbius function $$\mu(n)$$ is defined as follows: $\mu(1) = 1;$ If $$n > 1,$$ write $$n = p_1^{a_1} \dots p_k^{a_k}$$ (prime factorization). Then \begin{align*} \mu(n) & = (-1)^k \text{ if } a_1 = a_2 = \dots = a_k = 1, \\ \mu(n) & = 0 \text{ otherwise}. \end{align*} If $$n \ge 1,$$ we have $\sum_{d \mid n} \mu(d) = \begin{cases} 1 & \text{ if } n = 1, \\ 0 & \text{ if } n > 1. \end{cases}$

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 see soon in this post that the Euler's totient function can be expressed as a Dirichlet product of the Möbius function and the arithmetical function $$N(n) = n$$.

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 $$s = \sigma + it$$ is a complex number with its real part $$\sigma > 1$$, we have $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}.$ This immediately shows that $$\zeta(s) \ne 0$$ for $$\sigma > 1$$.

Dirichlet Product

If $$f$$ and $$g$$ are two arithmetical functions, their Dirichlet product $$f * g$$ is defined as: $(f * g)(n) = \sum_{d \mid n} f(d) g\left( \frac{n}{d} \right).$ Dirichlet products appear to pop up magically at various places in number theory. Here is a simple example: $\varphi(n) = \sum_{d \mid n} \mu(d) \frac{n}{d}.$ Therefore in the notation of Dirichlet products, the above equation can also be written as $\varphi = \mu * N$ where $$N$$ represents the arithmetical function $$N(n) = n$$ for all $$n$$.

Hurwitz Zeta Function

For complex numbers $$s = \sigma + it$$, the Hurwitz zeta function $$\zeta(s, a)$$ is initially defined for $$\sigma > 1$$ as $\zeta(s, a) = \sum_{n=0}^{\infty} \frac{1}{(n + a)^s}$ where $$a$$ is a fixed real number, $$0 < a < 1$$. Then by analytic continuation, it is defined for $$\sigma \le 1$$ as $\zeta(s, a) = \Gamma(1 - s)I(s, a)$ where $$\Gamma$$ represents the gamma function $\Gamma(s) = \int_0^{\infty} x^{s - 1} e^{-x} \, dx$ defined for $$\sigma > 0$$ and also defined, by analytic continuation, for $$\sigma \le 0$$ except for $$\sigma = 0, -1, -2, \dots$$ (the nonpositive integers) and $$I(s, a)$$ is defined by the contour integral $I(s, a) = \frac{1}{2\pi i} \int_C \frac{z^{s-1} e^{az}}{1 - e^z} \, dz$ where $$0 < a \le 1$$ and the contour $$C$$ is a loop around the negative real axis composed of three parts $$C_1$$, $$C_2$$, and $$C_3$$ such that for $$c < 2\pi$$, we have $$z = re^{-\pi i}$$ on $$C_1$$ and $$z = re^{\pi i}$$ on $$C_3$$ as $$r$$ varies from $$c$$ to $$+\infty$$, and $$z = ce^{i \theta}$$ on $$C_2$$, $$-\pi \le \theta \le \pi$$.

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 $\zeta(s) = \zeta(s, 1) = \sum_{n=1}^{\infty} \frac{1}{n^s}.$ Yes, the $$\zeta$$ symbol is overloaded: $$\zeta(s, a)$$ is the Hurwitz zeta function whereas $$\zeta(s)$$ is the Riemann zeta function. This relationship between the Riemann zeta function and the Hurwitz zeta function along with the analytic continuation of the Hurwitz zeta function opens new doors into the wonderful world of complex numbers and let us obtain beautiful and profound facts about the Riemann zeta function such as the fact that it has zeros at negative even integers, i.e., $$\zeta(n) = 0$$ for $$n = -2, -4, -6, \dots$$ and the fact that $$\zeta(0) = -\frac{1}{2}$$ and $$\zeta(-1) = -\frac{1}{12}$$ and so on.

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 $$\zeta(0) = -\frac{1}{2}$$ and $$\zeta(-1) = -\frac{1}{12}$$ using the analytic continuation of the Hurwitz zeta function today.