Proof of the Transcendence of the Euler–Mascheroni Constant γ

Qi Jin introduced in his paper a method for proving the irrationality of πe, π/e, π+e, and π−e. Building upon this approach, we present an optimized version of the method and use it to prove that the Euler–Mascheroni constant γ is transcendental, and hence irrational. Furthermore, we establish the transcendence of πe, π/e, π+e, and π−e. γ, πe, π/e, π+e, and π−e are both irrational and transcendental.