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Optimal Cosine Polynomials for Riemann Zeta Zero-Free Region

Hong Sheng Tan

Abstract

The zero-free region of the Riemann zeta function plays a pivotal role in understanding the distribution of prime numbers. Historically, E. Landau used a particular type of non-negative cosine polynomial to construct this region. Subsequent research by various mathematicians sought the optimal cosine polynomial that provides the largest zero-free region under this construction. In 1992, V. V. Arestov identified the optimal polynomial for all degrees up to 6. This article employs numerical methods to determine the optimal polynomials of degrees 7 and 8, as well as the corresponding zero-free region of the Riemann zeta function.

Optimal Cosine Polynomials for Riemann Zeta Zero-Free Region

Abstract

The zero-free region of the Riemann zeta function plays a pivotal role in understanding the distribution of prime numbers. Historically, E. Landau used a particular type of non-negative cosine polynomial to construct this region. Subsequent research by various mathematicians sought the optimal cosine polynomial that provides the largest zero-free region under this construction. In 1992, V. V. Arestov identified the optimal polynomial for all degrees up to 6. This article employs numerical methods to determine the optimal polynomials of degrees 7 and 8, as well as the corresponding zero-free region of the Riemann zeta function.

Paper Structure

This paper contains 22 sections, 32 theorems, 273 equations, 23 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The exact value of $V_2$ and $V_3$
  3. Some properties of linear functional $v(f)$.
  4. Exact value of $V_2$.
  5. Exact value of $V_3$.
  6. Methodology
  7. Karush-Kuhn-Tucker Method
  8. Penalty Function
  9. General Procedure
  10. Exact value of $V_4$.
  11. Revisit of Arestov and Kondrat'ev's method.
  12. Exact value of $V_4$.
  13. Exact value of $V_5$.
  14. Exact value of $V_5$.
  15. Exact value of $V_6$.
  16. ...and 7 more sections

Key Result

Theorem 2.1

Given let $m$ be the minimum of $f(\phi)$ when $\phi \in[0,2 \pi)$, then

Figures (23)

  • Figure 1: Graph of $v(f(\phi))$.
  • Figure 2: The lower bound of $\frac{\Delta_1}{\Delta_2}$.
  • Figure 3: Graph of $v(\bar{f} (\phi))$.
  • Figure 4: Graph of $v(f(\phi))$.
  • Figure 5: The ratio ${} \frac{p_{2}(a)}{(\sqrt{a}-1)^2}$.
  • ...and 18 more figures

Theorems & Definitions (57)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 47 more