I recently completed reading the book Introduction to Analytic Number Theory written by Tom M. Apostol and publised in 1976. It is a fantastic book that takes us through a breathtaking journey of analytic number theory. It begins with simple properties of divisibility and ends with integer partitions. During this journey, we learn about several fascinating concepts such as the Möbius function, Dirichlet multiplication, Chebyshev's functions, Dirichlet characters, quadratic residues, the Riemann zeta function, and the analytic proof of the prime number theorem.
One of the things about the book that caught my interest from the very beginning was its front cover. It has a peculiarly drawn grid of white boxes and red empty regions that looks quite interesting. Here is the grid from the front cover of the book:
Can we come up with a simple and elegant rule that defines this grid? Here is one I could come up with:
We define \( \gcd(x, y) \) to be a nonnegative common divisor of \( x \) and \( y \) such that every common divisor of \( x \) and \( y \) also divides \( \gcd(x, y) \). Let us now see if we can explain some of the interesting properties of this grid using the above rule:
When \( x = 0 \) and \( y \ne 1, \) we get \( \gcd(x, y) = \lvert y \rvert \ne 1, \) so the entire column at \( x = 0 \) has boxes except at \( (0, 1). \) Similarly, the entire row at \( y = 0 \) has boxes except at \( (1, 0). \)
The cell \( (0, 0) \) has a box because \( \gcd(0, 0) \ne 1 \). In fact, \( \gcd(0, 0) = 0. \) This follows from the definition of the \( \gcd \) function. We will discuss this in more detail later in this post.
Every diagonal cell \( (x, x) \) has a box except at \( (1, 1) \) because \( \gcd(x, x) = \lvert x \rvert \) for all integers \( x. \)
The grid is symmetric about the diagonal cells \( (x, x) \) because \( \gcd(x, y) = \gcd(y, x). \)
A column at \( x \) has exactly one cell below the diagonal if and only if \( x \) is prime. For example, check the column for \( x = 5. \) It has exactly one cell below the diagonal. We know that \( 5 \) is prime. Now check the column for \( x = 6. \) It has four cells below the diagonal. We know that \( 6 \) is not prime.
Let us now elaborate the second point in the list above. If \( \gcd(0, 0) \) is \( 0 \), then \( 0 \) must divide \( 0. \) Does \( 0 \) really divide \( 0? \) Isn't \( 0/0 \) undefined? Yes, even though \( 0/0 \) is undefined, \( 0 \) divides \( 0 \). We say an integer \( d \) divides an integer \( n \) when \( n = cd \) for some integer \( c. \) We have \( 0 = 0 \cdot 0, \) so indeed \( 0 \) divides \( 0 \).
We have shown that \( 0 \) divides \( 0 \) but we have not shown yet that \( \gcd(0, 0) = 0. \) Is \( \gcd(0, 0) \) really \( 0? \) Every integer divides \( 0 \), e.g., \( 1 \) divdes \( 0, \) \( 2 \) divides \( 0, \) \( 3 \) divides \( 0, \) etc. There does not seem to be a greatest common divisor of \( 0 \) and \( 0. \) Shouldn't \( \gcd(0, 0) \) be called either infinity or undefined? No, we need to look at the definition of \( \gcd \) introduced earlier. As per the definition, every common divisor of integers \( x \) and \( y \) must also divide \( \gcd(x, y). \) With this requirement in mind, we see that \( \gcd(0, 0) \) must be \( 0 \). This definition also makes \( \gcd(n, 0) = \gcd(0, n) = \lvert n \rvert \) for all integers \( n \). Further, this definition makes Bézout's identity hold for all integers. Bézout's identity states that there exists integers \( m \) and \( n \) such that \( mx + ny = \gcd(x, y). \) Indeed if we have \( \gcd(0, 0) = 0, \) we get \( 0 \cdot 0 + 0 \cdot 0 = 0 = \gcd(0, 0). \)
That's all I wanted to share about the front cover of the book. While the front cover is quite interesting, the content of the book is even more fascinating. I found chapters 12 and 13 of the book to be the most interesting. In chapter 12, the book teaches how to prove that the Riemann zeta function \( \zeta(s) \) vanishes at every negative even integer \( s \). Through several contour integrals and clever use of Cauchy's residue theorem, it shows in the end that \( \zeta(-2n) = 0 \) for \( n = 1, 2, 3, \dots. \) In chapter 13, the book shows us how to obtain zero-free regions where \( \zeta(s) \) does not vanish. The book exposes various subtle nuances of the zeta function with great rigour and thoroughness. Results like \( \zeta(-1) = 1/12 \) that once felt mysterious look crystal clear and obvious after working through this book. I strongly recommend this book to anyone who wants to learn analytic number theory.