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On the exact location of the non-trivial zeros of Riemann's zeta function

Juan Arias de Reyna, Jan van de Lune

Abstract

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.

On the exact location of the non-trivial zeros of Riemann's zeta function

Abstract

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.

Paper Structure

This paper contains 10 sections, 36 theorems, 145 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Phase and argument of a function.
  3. The function $\vartheta(t)$.
  4. The function $\kappa(t)$.
  5. Hypothesis P and its consequences.
  6. Connection of $\kappa'(t)$ with the zeros of $\zeta'(s)$.
  7. Counting the zeros of $\zeta(s)$.
  8. Connections between the zeros of $\zeta(s)$ and $\zeta'(s)$.
  9. The functions $E(t)$ and $S(t)$.
  10. Extension to other $L$-functions.

Key Result

Proposition 2

If $f\colon \mathbf{R}\to\mathbf{C}\smallsetminus \{0\}$ is a real analytic function, then there exists a real analytic function $g$ such that $f(t)=e^{g(t)}$ for every $t\in\mathbf{R}$.

Figures (8)

  • Figure 1: $\mathop{\mathrm{ph}}\nolimits\zeta'({ \frac{1}{2}}\,+it)$
  • Figure 2: $\kappa(t)$
  • Figure 3: $Z(t)$ near the origin.
  • Figure 4: $\kappa'(t)$
  • Figure 5: $\kappa(t)$
  • ...and 3 more figures

Theorems & Definitions (80)

  • Definition 1
  • Proposition 2
  • proof
  • Corollary 3
  • Proposition 4
  • proof
  • Definition 5
  • Example 1
  • Example 2
  • Example 3
  • ...and 70 more