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Explicit Bound of $\pmb{|ζ\left(1+it\right)|}$

Eunice Hoo Qingyi, Lee-Peng Teo

TL;DR

This work derives explicit upper bounds for $|ζ(1+it)|$ on the line $s=1+it$. It combines a refined Euler–Maclaurin representation with a carefully controlled finite-sum approximation to obtain a sharp bound $|ζ(1+it)| \le 0.6443\log t$ for all $t\ge e$, with equality near $t=17.7477$, and a tighter bound $|ζ(1+it)| \le \tfrac{1}{2}\log t+0.6633$ for $t\ge e$ that becomes the better choice when $t\ge 100$. The Riemann–Siegel formula is then employed to extend reliable bounds to larger $t$, yielding $|ζ(1+it)| \le \tfrac{1}{2}\log t+0.6633$ for all $t\ge e$ and a sharper $0.5480\log t$ bound for $t\ge 652.3704$. The results also establish the optimality of the $0.6443$ constant among bounds of the form $v\log t$ and demonstrate the practical limitations of exponential-sums-based approaches in beating the half-log growth, while highlighting the RS method’s utility for large $t$. Overall, the paper provides explicit, verifiable constants and situates the new bounds within the landscape of classical results.

Abstract

In this work, we show that for all $t\geq e$, \[|ζ(1+it)|\leq 0.6443 \log t. \] The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that \[|ζ(1+it)|\leq\frac{1}{2}\log t+0.6633\hspace{1cm}\text{when}\;t\geq e.\]When $t\geq 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$.

Explicit Bound of $\pmb{|ζ\left(1+it\right)|}$

TL;DR

This work derives explicit upper bounds for on the line . It combines a refined Euler–Maclaurin representation with a carefully controlled finite-sum approximation to obtain a sharp bound for all , with equality near , and a tighter bound for that becomes the better choice when . The Riemann–Siegel formula is then employed to extend reliable bounds to larger , yielding for all and a sharper bound for . The results also establish the optimality of the constant among bounds of the form and demonstrate the practical limitations of exponential-sums-based approaches in beating the half-log growth, while highlighting the RS method’s utility for large . Overall, the paper provides explicit, verifiable constants and situates the new bounds within the landscape of classical results.

Abstract

In this work, we show that for all , The equality is achieved when . We also use the Riemann-Siegel formula and numerical computations to show that When , the bound is better than the bound .

Paper Structure

This paper contains 5 sections, 15 theorems, 208 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Classical Representation of the Riemann Zeta Function
  3. Bounds of $|\zeta(1+it)|$ Using Exponential Sums
  4. Bounds of $|\zeta(1+it)|$ Using the Riemann--Siegel Formula
  5. Concluding the Proof of Theorem \ref{['maintheorem']}

Key Result

Theorem 1.1

For $t\geq e$, we have and The result 20250612_7 is the best possible for a bound of the form $|\zeta(1+it)|\leq v\log t$ that holds for all $t\geq e$. When $t\geq 100$, the bound 20250612_8 is better than the bound 20250612_7. When $t\geq 652.3704$, we can obtain a better bound of the form $|\zeta(1+it)|\leq v\log t$, which When $652.3704 \leq t\leq 10^6$, the bound 20250613_7 is better than t

Figures (5)

  • Figure 1: The function $|C_0(p)|$ when $0\leq p\leq 1$.
  • Figure 2: The function $|C_1(p)|$ when $\sigma=0$ and $0\leq p\leq 1$.
  • Figure 3: The function $|C_1(p)|$ when $\sigma=1$ and $0\leq p\leq 1$.
  • Figure 4: The figures show comparisons of $|\zeta(1+it)|$ with $\frac{1}{2}\log t+0.6633$ when $e\leq t\leq 10^6$ and when $e\leq t\leq 500$.
  • Figure 5: The figures show $|\zeta(1+it)|/\log t$ when $e\leq t\leq 100$ and when $10\leq t\leq 20$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • ...and 13 more