Dirichlet's eta and beta functions: Concavity and fast computation of their derivatives

Abstract

For any a∈(0,∞), we prove the strict concavity of the functionηa(t):=∑m=0∞(−1)m(am+1)t on (0,∞), and provide fast computations of their derivatives on (0,∞). We give short proofs mainly based on differentiation formulas concerning the gamma process. In particular, our results apply to Dirichlet's eta and beta functions η1(t) and η2(t), respectively.