Why we shouldn't believe that $\pi(x)$ has connection to the constant $\pi$?

Question

It is well known that $\pi(x)$ is the prime counting function , This notation was introduced by number theorist Edmund Landau in 1909 if we define this function as:

$\pi:\mathbb{Z}\to \mathbb{Z} $ then we have the application is $\pi$ which it is identical to the constant $\pi=3.14\dotso$. Why do number theorists define this function using the constant $\pi$? Is there any connection? and how they weren't afraid about confusing between prime counting function and $\pi$ constant?

Answer 1

There was no fear of confusion for the same reason that $i=\sqrt {-1}$ is rarely confused with an index like $$\sum_{i=1}^\infty{x^i}$$ It's because they appear in different contexts. It's not true that $\pi$ never shows up in number theory, but if the constant $\pi$ is intended it would usually not have parentheses after as it would as the prime counting function. If there were potential for confusion, most authors would contort their notation to avoid this.

Answer 2

You are right, and this concern about frozen notation was addressed by Paul R. Halmos in his famous article How to write mathematics:

As history progresses, more and more symbols get frozen. The standard examples are $e$, $i$ and $\pi$, and, of course, $0,1,2,3, \ldots$ (Who would dare write “Let $6$ be a group.”?) A few other letters are almost frozen: many readers would fell offended if $n$ were used for a complex number, $\epsilon$ for a positive integer, and $z$ for a topological space. (A mathematician’s nightmare is a sequence $n\varepsilon$ that tends to $0$ as $\varepsilon$ becomes infinite.)

This is why Halmos recommends against freezing new symbols. Now, regarding your question about $\pi(x)$, it was observed in the comments that Landau's notation applies to a function, but it was also observed that $\pi(x+1)$ is still be ambiguous. Thus the risk exists but hopefully the context should be sufficient to avoid any confusion.

Computer scientists also face this problem but solve it in two different ways. First, by using types: the constant $\pi$ (of type real number) is a different object than the function $\pi:{\Bbb Z} \to {\Bbb Z}$. Secondly, by using local variables: the scope of a local variable is limited to the function that contains its definition and hence the same letter can be reused for another local variable. This is exactly what we do when we reuse $n$ in $\sum_n x_n$ and $\sum_n y_n$. Extrapolating on this notion, Landau's notation can be seen as local to number theory.