Continuing Problem 11, show that (1 s) is an analytic function of the complex variable s

Continuing Problem 11, show that Γ(1 − s) is an analytic function of the complex variable s for |s| sufficiently small and that the convergence in equation (14.9) is uniform. Consequently, the moments of λMn − ln n converge to the moments of the extreme value density e−e−u e−u. Prove that this density has mean and variance d

ds ln Γ(1 − s)|s=0 = γ

d2 ds2 ln Γ(1 − s)|s=0 = ∞

k=1 1

k2

= π2 6 , where γ is Euler’s constant. (Hint: Quote whatever facts you need about the log gamma function ln Γ(t) [12].)