Grant's masterful video https://www.youtube.com/watch?v=NaL_Cb42WyY&ab_channel=3Blue1Brown ("Pi hiding in prime regularities") describes a way of computing $\pi$ that ultimately leads us to the development of Dirichlet characters, and more generally the beginnings of analytic and algebraic number theory. I find the video to be mostly straightforward (brilliant, but not to an eye-watering degree), at least until minute 20:44.
There, in a stroke of genius so immense that I can only stare in wonder, we are introduced to a deceptively simple function $\chi$, that ends up giving us a very illuminating form of describing the number of lattice points on a circle of radius $\sqrt N$.
My inquiry is this: let's say I was trying to come up with this formula for $\pi$ on my own. I play around with squares, finding Fermat's Christmas theorem, and its relation to factorization in the Gaussian integers. And I find the preliminary formula for $\#_N :=$ the number of lattice points on a circle of radius $\sqrt N$, namely: breaking down $N$ into prime factors $p^e$ (prime $p$, exponent $e$) and building the desired quantity $\#_N$ by multiplying by [$1$ if $p \equiv 3 \pmod 4$ and $e$ is even], [$0$ if $p \equiv 3 \pmod 4$ and $e$ is odd], and [$(e+1)$ if $p \equiv 1 \pmod 4$] for each prime factor $p$, and finally multiplying by $4$.
Is there then some kind of technique, general philosophy/problem-solving strategy, or thought process I could go through to realize that I could define this function $\chi$ (multiplicative, cyclic) that would simplify this formula into something more illuminating/less "awkward"? I mean, writing it out in this form involving sums $\chi(p^0) + \chi(p^1) + \ldots + \chi(p^e)$ at first looks so much more complicated, but is actually so much simpler, I just don't know how someone would be able to see past the "complicated" to the "simpler" a priori.
It all seems so...magical and almost coincidental that such a manipulation could be pulled off.
One approach to answering this question would be to imagine you were writing an introductory textbook to the subject, and wanted to insert this problem in the text. What topics/problems would you write about before this one so as to make this problem and its solution appear in a "more natural" context?
Short answer: when dealing with a function $f(n)$ that is multiplicative/related to factorization, it is standard (once one has experience in this area!) to look at the functions $g(n)$ and $h(n)$ that satisfy $h(n) = \sum_{d\mid n} f(d)$ and $f(n) = \sum_{d\mid n} g(d)$ (both sums are over positive integers $d$ that divide $n$).
In this context, $\#_N = \sum_{d\mid n} \chi(d)$, which motivates looking at the function $\chi$.