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We generalize Sondow's (ir)rationality criteria for Euler's
constant and give necessary and sufficient conditions for
irrationality of generalized Euler's constant $\gamma_\alpha$, as
well as we obtain new asymptotic formulas for computing
$\gamma_\alpha$. The proof is based on constructing linear forms
in 1,$\gamma_\alpha$ and logarithms of rational numbers.
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