Riccati--Gamma Dynamics for Concavity and Asymptotics of Generalized Dirichlet Eta Functions

Abstract

We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\eta_{a}(t)=\sum_{m\ge0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which includes the classical Dirichlet eta and beta functions. Using a Mellin--Laplace representation of $\eta_{a}$ as $\mathbb{E}[f_{a}(X_{t})]$, where $f_{a}$ is a scaled logistic function and $(X_{t})$ a standard Gamma process, we show that the logarithmic derivative $\varphi_{a}(t)=\eta_{a}'(t)/\eta_{a}(t)$ satisfies a non-homogeneous Riccati equation with strictly negative forcing. This single inequality yields strict concavity and strict log-concavity of $\eta_{a}$, positivity and monotonicity of $\varphi_{a}$, and the precise asymptotic law $\varphi_{a}(t)=\log(a+1)(a+1)^{-t}+O((a+2)^{-t})$. We further prove that $\varphi_{a}(t)/\varphi_{a,e}(t)\to 2/\log(a+1)$ as $t\to\infty$, where $\varphi_{a,e}(t)=-\eta_{a}''(t)/(2\eta_{a}(t))$, obtaining in particular the trapping inequality $0<\varphi_{a,e}(t)<\varphi_{a}(t)$ for all sufficiently large $t$ when $a<e^{2}-1$. We also present a self-contained geometric-rate algorithm (rate $1/3$) for computing $\eta_{a}^{(k)}(t)$ together with a sharp error bound. High-precision numerical experiments confirm all results. As an application, we show that the Riccati--Gamma dynamics of $\eta_{a}$ and $\varphi_{a}$ provide a principled mechanism for musical synthesis, generating a complete melody whose pitch and rhythm are governed by these functions.