A simple inequality relating the Euler-Riemann zeta function, digamma, and cotangent over the unit interval
Michael Andrew Henry
TL;DR
The paper proves the inequality $\pi \cot \pi x < \zeta(x) - \psi(x)$ for $0 < x < 1$, by reducing it to the classical identity $\psi(1-x)-\psi(x)=\pi \cot \pi x$ and showing $\psi(1-x) < \zeta(x)$. It constructs a simple, linear comparator $f(x)=b x+\tfrac{1}{2}$ with $b=\gamma-\tfrac{1}{2}$ and proves $f(x) \le \zeta(x) + \dfrac{1}{1-x}$ on $(0,1)$ by leveraging a monotonicity property of $\zeta(x)+\dfrac{1}{1-x}$, yielding the main inequality. The work also proposes a conjectured strengthening with a Kubert-Knuth replication viewpoint, and discusses potential connections to the Riemann hypothesis by reducing questions about RH to the unit interval behavior of $\zeta(x)$ on $(0,1)$. These results tie together the zeta function, the digamma function, and the cotangent function through an elementary, self-contained argument and a framework for sharper bounds.
Abstract
We prove an inequality featuring three well-known functions from analysis, namely the cotangent, the Euler-Riemann zeta function, and the digamma function. Aside from a simple proof of our result, we give a conjectured strengthening. We offer various remarks about the origins of this problem.