On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant

Tanguy Rivoal
2012 The Michigan mathematical journal  
We prove new results concerning the arithmetic nature values of the Gamma function Γ at algebraic points and Euler's constant γ. We prove that for any α ∈ Q\Z, α > 0, at least one of the numbers Γ is an irrational number. Similarly, at least one of the numbers γ = − ∞ 0 log(t)e −t dt and Gompertz's constant ∞ 0 e −t /(1 + t)dt is an irrational number. Quantitative statements, obtained by means of Nesterenko's linear independence criterion, strengthen these irrationality assertions. G α (z) := z
more » ... tions. G α (z) := z −α ∞ 0 (t + z) α−1 e −t dt.
doi:10.1307/mmj/1339011525 fatcat:u67stsevkreq3hweh67ilgatwa