What does this double sided arrow $\longleftrightarrow$ mean? What is $\longleftrightarrow$ used for in mathematics? I know about $\iff$ being used for "If and only if". Are they the same thing? I was watching a YouTube video that said:
$$\sum^{\infty}_{n=1} {1\over n^x} \longleftrightarrow \int^{\infty}_{1} {1\over t^x} dt$$
The teacher mentions convergence/divergence, but I was confused when the notation came up.

On the one hand, $\longleftrightarrow $ is used for connecting propositional formulas (e.g. $p\to q \lor (p\longleftrightarrow q) \land \lnot w$). You can understand it as a binary operator like AND or OR, which are represented by $\land $ and $\lor $ symbols, as you would know.
Here you can see its truth table.
$$\begin{array}{|c|c|c|}
\hline
p&q&p\longleftrightarrow q\\
\hline
T&T&T\\
\hline
T&F&F\\
\hline
F&T&F\\
\hline
F&F&T\\
\hline
\end{array}$$
On the other hand, $\iff $ is used as a connective of propositional formulas. You can see both uses here:
$$p\longleftrightarrow q \iff (p\to q) \land (q \to p)$$
And what does $a \iff b $ means? If you write $a\iff b $, then you could actually say the same by writing down that the bicondition $a \text { is true} \longleftrightarrow b \text{ is true} $ is always true. Note that this works whatever the truth values of $a \text { is true} $ or $b \text { is true}$ are.
Edit: in another fields a part of logic, (at least in basic degrees), choosing one or the other does not matter too much ($\longleftrightarrow $ or $\iff $ are just "lazy" math translations of simple English connector "if and only if").
