Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region

TL;DR

The paper develops an explicit kth derivative test via van der Corput's A^{k−2}B method to bound exponential sums uniformly for all k ≥ 4, yielding explicit bounds for ζ(s) on vertical lines σ_k = 1 − k/(2^k − 2) in the critical strip: |ζ(σ_k + it)| ≤ 1.546 t^{1/(2^k−2)} log t for t ≥ 3. These bounds are then used to establish an explicit zero-free region σ > 1 − log log|t| /(21.233 log|t|) for exp(170) ≤ t ≤ exp(532141), representing the largest known region in that t-range; the corollary extends the result to other ranges via a Ford–Richert style argument with a degree-46 trigonometric polynomial. The main contribution is an explicit, uniform kth-derivative framework that sharpens zeta-s bounds beyond convexity-based methods and yields practically computable zero-free regions, with clear pathways for further refinement of constants and phase choices.

Abstract

We derive explicit upper bounds for the Riemann zeta-function $ζ(σ+ it)$ on the lines $σ= 1 - k/(2^k - 2)$ for integer $k \ge 4$. This is used to show that the zeta-function has no zeroes in the region $$σ> 1 - \frac{\log\log|t|}{21.233\log|t|},\qquad |t| \ge 3.$$ This is the largest known zero-free region for $\exp(171) \le t \le \exp(5.3\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1: Second-derivative test
• proof
• Remark
• Lemma 2.2
• proof
• Lemma 2.3: Explicit $A$ process
• proof
• Lemma 2.4: Explicit third derivative test