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

TL;DR

This paper introduces a unified dynamical framework for analyzing generalized Dirichlet eta functions, revealing their concavity, asymptotics, and derivative properties through Riccati equations and Gamma process representations.

Contribution

It develops a novel Riccati--Gamma dynamics approach to study the qualitative behavior and asymptotics of generalized Dirichlet eta functions, including new inequalities and a geometric-rate derivative algorithm.

Findings

Proves strict concavity and log-concavity of eta functions on (0,∞).

Derives exact asymptotic ratios and inequalities for derivatives of eta functions.

Provides a high-precision, rate 1/3 algorithm for derivatives with error bounds.

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 specialises to the classical Dirichlet eta and beta functions for $a=1$ and $a=2$. Building on a Mellin--Laplace representation of $\eta_{a}$ as the expectation $\mathbb{E}[f_{a}(X_{t})]$ of a scaled logistic function evaluated along a standard Gamma process $(X_{t})_{t\ge 0}$, we prove that the logarithmic derivative $\varphi_{a}(t)=\eta_{a}^{\prime }(t)/\eta_{a}(t)$ satisfies a non\-homogeneous Riccati equation whose forcing term is strictly negative on $(0,\infty)$. This single dynamical inequality yields, in one step, the strict concavity and strict log-concavity of $\eta_{a}$ on $% (0,\infty)$, the positivity and monotonicity of $\varphi_{a}$, and the exact leading-order expansion $\varphi_{a}(t)=\log(a+1)(a+1)^{-t}+O((a+2)^{-t})$ as $t\to\infty$. We further compute the exact asymptotic ratio $% \varphi_{a}(t)/\varphi_{a,e}(t)\to 2/\log(a+1)$ as $t\to\infty$, where $% \varphi_{a,e}(t):=-\eta_{a}^{\prime \prime }(t)/(2\eta_{a}(t))$; in particular, for $a<e^{2}-1$ (which covers the classical Dirichlet eta and beta cases $a=1,\,2$) this ratio exceeds one, yielding the trapping inequality $0<\varphi_{a,e}(t)<\varphi_{a}(t)$ on a half-line $% (T_{*},\infty) $, which is equivalent to the structural curvature inequality $\eta_{a}^{\prime \prime }(t)+2\eta_{a}^{\prime }(t)>0$. Finally, we present a self-contained, rigorously justified geometric-rate algorithm (rate $1/3$) for the $k$-th derivative of $\eta_{a}$, together with a sharp combinatorial error bound. High-precision numerical experiments confirm every theoretical statement.