Inequalities for $\zeta(s)-\psi(1-s)$ related to a conjecture of Henry

Liwen Gao; Xuejun Guo

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arXiv · 2601.09276

Inequalities for $\zeta(s)-\psi(1-s)$ related to a conjecture of Henry

Liwen Gao·Xuejun Guo

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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 $\zeta(s)-\psi(1-s)$ 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 $\pi$, $\log (2\pi)$ and Stieltjes constants. These results lead to new inequalities for $\zeta(s)-\psi(1-s)$ and shed further light on the conjecture.

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Liwen Gao, Xuejun Guo. 2026-01-14. Inequalities for $\zeta(s)-\psi(1-s)$ related to a conjecture of Henry. https://arxiv.org/abs/2601.09276

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