GCD Grid
I recently completed reading the book Introduction to Analytic Number Theory written by Tom M. Apostol and published in 1976. It is a fantastic book that takes us through a breathtaking journey of analytic number theory. The journey 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, etc. An analytic proof of the prime number theorem is also presented in the book.
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
-
When
and we get so the entire column at has boxes except at Similarly, the entire row at has boxes except at -
The cell
has a box because In fact, This follows from the definition of the function. We will discuss this in more detail later in this post. -
Every diagonal cell
has a box except at because for all integers -
The grid is symmetric about the diagonal cells
because -
A column at
has exactly one cell below the diagonal if and only if is prime. For example, check the column for It has exactly one cell below the diagonal. We know that is prime. Now check the column for It has four cells below the diagonal. We know that is not prime.
Let us now elaborate the second point in the list above. If
We have shown that
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