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 ${\eta }_a(t)={\sum }_{m\ge 0}(-1{)}^m(am+1{)}^{-t}$, $a>0$, $t>0$, 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 ${\phi }_a(t)={\eta }_a^{\prime }(t)/{\eta }_a(t)$ 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 ${\phi }_a$, and the precise asymptotic law ${\phi }_a(t)=\log (a+1)(a+1{)}^{-t}+O((a+2{)}^{-t})$. We further prove that ${\phi }_a(t)/{\phi }_{a,e}(t)\to 2/\log (a+1)$ as $t\to \infty$, where ${\phi }_{a,e}(t)=-{\eta }_a^{\prime \prime }(t)/(2{\eta }_a(t))$, obtaining in particular the trapping inequality $0<{\phi }_{a,e}(t)<{\phi }_a(t)$ 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 ${\phi }_a$ provide a principled mechanism for musical synthesis, generating a complete melody whose pitch and rhythm are governed by these functions.

24 pages