Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$?
Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and $\zeta(\frac{141}{23}) \approx e^\frac{1}{64}$.  I also figured out that $\zeta(x)$ approaches $e^{2^{-x}}$ but I'm not sure that helps explain why these almost-equalities exist.  How to quantify how surprising these almost-equalities are, and what is the explanation for them if any?
EDIT: There does seem to be a pattern here:
$\log \zeta(n + (\frac{2}{3})^{n-1}) \approx 2^{-n}$ for $n = 1,2,3,4,...$.  I think this formula explains the observations but where does it come from?
BONUS, since I've retagged this as a soft-question already: Is there any wrong but somehow plausible argument that two random integers are relatively prime with probability $\frac{1}{\sqrt{e}}$?  I guess it would be like a Lucky Larry story.
 A: Here is one way Lucky Larry might figure the limit probability that a number is squarefree: Larry already knows that $\zeta(2) \le 2 = 1+\sum_{1 \le n} \frac{1}{n \cdot (n+1)}$, which means no more than than half of all integers are squareful. Let $F_{n}:\{1,2,...,n\}\rightarrow\{1,2,...,2 \cdot n\}$ be a randomly chosen function whose range consists of all squareful integers between $1$ and $2 \cdot n$.  So the condition of being squarefree is equivalent to not being in the range, and since the function is randomly chosen, the probability is $(1-\frac{1}{2 \cdot n})^n \rightarrow \frac{1}{\sqrt{e}}$.  But of course the correct answer is $\frac{6}{\pi^2}$.
The problem with the above is that it ignores the constraint "whose range consists of all squareful numbers between $1$ and $2 \cdot n$" and the formula used only applies when the function is uniformly chosen.  The almost-equality shows that this isn't always as big a mistake as it may seem.
A: $\zeta(2)$ is almost equal to $\sqrt e$ because if it weren't you'd be asking why it's almost equal to $\log_{10}44$. Honestly, interesting numbers are $\epsilon$-dense in the reals, where $\epsilon$ depends on what you find interesting, so it's guaranteed there will be interesting numbers close to each other, for no deeper reason at all. 
