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1

Well, to get a good approximation for $\pi$, say, you only need to measure a rope around a circle or compute a well-known series up to sufficiently many summands. The Landau constants, on the other hand, make a statement about a very large family of vary much varying functions ...

3

It comes up in weather dynamics models where volume or surface area of spheres are relevant. It's not really anything special, it's not like you can read off weather patterns from its digits.

3

$\pi$, $e$, $\gamma$ (Euler's Constant) stand out as some of the more important mathematical constants. If we wanted we could declare any real number as a mathematical constant". However, the numbers that bear that name have been found useful in some context, and most commonly they have been found useful in many contexts. So the answer to your second ...

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