Euler-Mascheroni constant

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Recomm. for undergrads: no
Posted by: Juggernaut
on: July 19th, 2011
Question   Is Euler-Mascheroni constant an transcendental number?

Let $ \gamma:=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^n\left(\frac{1}{k}\right)-\ln(n)\right) $. The number $ \gamma $ has not been proved algebraic or transcendental. In fact, it is not even known whether $ \gamma $ is irrational.