A new improved explicit estimate for $ζ\left( 1/2+it\right)$

TL;DR

This work sharpens explicit subconvexity bounds for the Riemann zeta function on the critical line by combining an explicit van der Corput framework with extensive computations to control intermediate and bottleneck regions. The authors establish $| ext{zeta}(1/2+it)|\le 0.611\,t^{1/6}\log t$ for $t\ge 3$ and further improve to $| ext{zeta}(1/2+it)|\le 0.566\,t^{1/6}\log t$ for $t\ge 8.97\times 10^{17}$, while providing detailed proofs and computational validation that address gaps in prior work. Central to the method are refined second-derivative bounds, Kusmin–Landau-type estimates, and a meticulously organized bottleneck-region analysis spanning ten subintervals. The results have practical significance for explicit zero-free regions, zero-density estimates, and numerical checks of zeta-values, and the paper also discusses potential theoretical and computational improvements and their feasibility.

Abstract

In this paper, we present an improved explicit subconvexity result for the Riemann zeta function $ζ\left( s\right)$ along the critical line $s=1/2+it$, given by Hiary, Patel and Yang in 2024. This new bound is derived by combining a refined, explicit version of the van der Corput method together with computational calculations.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 2.1: Computational Bound
• proof
• Lemma 2.2: Riemann-Siegel Bound
• proof
• Lemma 2.3: Refined Kusmin-Landau Bound
• proof
• Remark 2.1
• Remark 2.2
• Lemma 2.4: Weyl differentiation