What is the best notation to write ``equations with reasons''?

Question

I'm learning my analytic number theory course. When I make notes, I find that it is effective to use ``equations with reasons'' as follow $$\begin{array}{rll} \vartheta(x)&\displaystyle=\sum_{p\leq x} \log p& \textrm{partial sum $\begin{cases} 1 & \textrm{$n$ is prime}\\ 0 & \textrm{otherwise} \end{cases}$ and $\log x$}\\ &\displaystyle =\pi(x)\log x-\int_1^x \frac{\pi(t)}{t}\mathrm{d} t & \textrm{Assume that $\displaystyle\pi(x)=\frac{x}{\log x}+O\left(\frac{x}{\log x}\right)$}\\ &\displaystyle=x-\int_1^x \frac{1}{\log t}\mathrm{d} t+O\left(x-\int_2^x \frac{1}{\log t}\mathrm{d} t\right) & \displaystyle\because\int_2^x \frac{1}{\log t}\mathrm{d} t\leq \frac{x}{\log x}\leq x \\ &\displaystyle=x+O(x)\\ \end{array}$$ My question is are there any standard on this notations? For example, the reason should put into which line, before the result of the result? And, is it acceptable to use the symbol $\because$?

Answer 1

This is a method I learned from a post by Asaf Karagila. \begin{align} A + B &= \quad \text{A short good reason} \\ C + D &= \quad \text{A much longer reason that takes multiple} \\ &\hspace{1.5em} \quad \text{lines because it is so much more complicated} \notag \\ &\hspace{1.5em} \quad \text{than the short reasons} \notag \\ E + F\text{.} \end{align} The idea is that the explanation says why you can go from one line to the next line.

Answer 2

Here are two examples of the notation I use, although I don't think there is just one way to do it.

$$\begin{align*} c^2 & = a^2 + b^2 & \text{(by Pythagorean Theorem)} \\ & = 2a^2 & \text{(since } a = b \text{)} \\ & = 2 & \text{(since } a = 1 \text{)} \end{align*}$$

and it's code:

\begin{align*}
    c^2 & = a^2 + b^2 & \text{(by Pythagorean Theorem)} \\
    & = 2a^2 & \text{(since } a = b \text{)} \\
    & = 2 & \text{(since } a = 1 \text{)}
\end{align*}

And the other example:

$$\begin{align*} c^2 = a^2 + b^2 & \Rightarrow c^4 = (a^2 + b^2)^2 & \text{(square both sides)} \\ & \Rightarrow c^4 = a^4 + 2a^2b^2 + b^4 & \\ & \Rightarrow c^4 = 2a^4 + 2a^2a^2 & \text{(since } a = b \text{)} \\ & \Rightarrow c^4 = 4a^4 & \\ & \Rightarrow a^4 = \frac{c^4}{4} & \\ \end{align*}$$

and it's code:

\begin{align*}
    c^2 = a^2 + b^2 & \Rightarrow c^4 = (a^2 + b^2)^2 & \text{(square both sides)} \\
    & \Rightarrow c^4 = a^4 + 2a^2b^2 + b^4 & \\
    & \Rightarrow c^4 = 2a^4 + 2a^2a^2 & \text{(since } a = b \text{)} \\
    & \Rightarrow c^4 = 4a^4 &  \\
    & \Rightarrow a^4 = \frac{c^4}{4} &  \\
\end{align*}