Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounding the integral of the argument of the Riemann Zeta function

Victor Amberger

TL;DR

The paper tackles the problem of obtaining explicit, tight bounds for the integral of the argument of the Riemann zeta function, $S_1(t)=\int_0^t S(u)\,du$, to strengthen Turing's method for verifying zeros of $\zeta(s)$. It refines the Brent-Lehman framework by incorporating the real part of the second logarithmic derivative of $\zeta(s)$ and leveraging modern bounds on $|\zeta(s)|$ via the Phragm\'en-Lindelöf principle, culminating in an explicit bound of the form $|S_1(t_2)-S_1(t_1)|\le a+b\log\log t_2+c\log t_2$ with optimally chosen constants around $t\sim 10^{12}$. A corrected version of the Gram-block theorem is established, and detailed computations produce admissible parameter sets and tables, demonstrating improved tightness over prior work (e.g., Trudgian). The results enable more efficient and reliable zero verification via Turing's method and are extendable to other $L$-functions through the same analytical framework.

Abstract

This article improves the estimate of $|S_1(t_2)-S_1(t_1)|$, which is the definite integral of the argument of the Riemann zeta-function between $t_1$ and $t_2$. Estimates of this quantity are needed to apply Turing's method to compute the exact number of zeta zeros up to a given height.

Bounding the integral of the argument of the Riemann Zeta function

TL;DR

The paper tackles the problem of obtaining explicit, tight bounds for the integral of the argument of the Riemann zeta function, , to strengthen Turing's method for verifying zeros of . It refines the Brent-Lehman framework by incorporating the real part of the second logarithmic derivative of and leveraging modern bounds on via the Phragm\'en-Lindelöf principle, culminating in an explicit bound of the form with optimally chosen constants around . A corrected version of the Gram-block theorem is established, and detailed computations produce admissible parameter sets and tables, demonstrating improved tightness over prior work (e.g., Trudgian). The results enable more efficient and reliable zero verification via Turing's method and are extendable to other -functions through the same analytical framework.

Abstract

This article improves the estimate of , which is the definite integral of the argument of the Riemann zeta-function between and . Estimates of this quantity are needed to apply Turing's method to compute the exact number of zeta zeros up to a given height.
Paper Structure (6 sections, 10 theorems, 72 equations, 2 tables)

This paper contains 6 sections, 10 theorems, 72 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Proof of Theorem \ref{['theorem abc loglog t']}
  4. Estimates for $S_1(t)$
  5. Computations and comparison to previous estimates
  6. Conclusion

Key Result

Theorem 1.1

Let $a$ and $b$ be positive real constants that satisfy for all $t_2>t_1\geq t_0$. Furthermore, let $K$ consecutive Gram Blocks with union $[g_n,g_p)$ satisfy Rosser's rule, such that $g_n\geq t_0$ and then

Theorems & Definitions (21)

  • Theorem 1.1: Brent-LehmanBrent
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 3.1: Lehman
  • proof : Proof of Theorem \ref{['theorem abc loglog t']}
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 11 more