Inequalities for $ζ(s)-ψ(1-s)$ related to a conjecture of Henry
Liwen Gao, Xuejun Guo
TL;DR
Problem and method: The paper investigates analytic inequalities for $F(s)=\zeta(s)-\psi(1-s)$, inspired by Henry's conjecture, by treating it as a single analytic object and applying Stieltjes expansions and precise integral representations. Main results: it proves $F''(s)>0$ and $F'(s)>0$ on $(0,1)$, derives explicit boundary limits $\lim_{s\to0^+}F'(s)=\frac{\pi^2}{6}-\frac{1}{2}\log(2\pi)$ and $\lim_{s\to1^-}F'(s)=\frac{\pi^2}{6}-\gamma_1$, and obtains the sharp linear bound $0<s< F(s)< b's+b$ with $b'=\gamma_0+\frac{1}{2}$ and $b=\gamma_0-\frac{1}{2}$. Methods are based on Dirichlet-series/digamma representations and uniform convergence arguments, yielding explicit constants involving $\pi$, $\log(2\pi)$ and the first Stieltjes constant. Implications: these inequalities illuminate Henry's conjecture, connect to fundamental constants, and suggest a path to extending similar bounds to the Hurwitz zeta function.
Abstract
In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $ζ(s)-ψ(1-s)$ as a unified analytic object, we establish its strict convexity and monotonicity on suitable intervals. Moreover, we obtain explicit boundary limits of the derivative, expressed in terms of $π$, $\log (2π)$ and Stieltjes constants. These results lead to new inequalities for $ζ(s)-ψ(1-s)$ and shed further light on the conjecture.