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Level curves for Zhang's Eta Function

Jeffrey Stopple

TL;DR

This work introduces a level-curve framework to classify zeros of the Riemann zeta function and its derivative via η(s)=π^{−s/2}Γ(s/2)ζ′(s). Under the Riemann Hypothesis with simple zeros, zeros of ζ′(s) are partitioned into three types (0,1,2) based on how the Re(η)=0 level curves exit toward the critical line, and a canonical mapping ties these to zeros of ζ(s), with N2(T)=N2′(T) and N1(T)=N1′(T). The paper provides asymptotic count relations, extensive computational data for ~1e6 zeros near T=10^{10}, and curvature-based arguments that connect liminf behaviors of (β′−1/2)log γ′ and (γ^+−γ^−)log γ′ for type-2 zeros, reducing a conjecture to a curvature bound on the level curve. An explicit Marden-type decomposition and a curvature identity show how the positions of all auxiliary zeros influence the curvature at each type-2 zero. An appendix extends the approach to random matrix analogs, drawing parallel results for zeros of characteristic polynomials of unitary matrices and their derivatives, reinforcing the deep structural similarity between zeta- and matrix-analytic spectra.

Abstract

Study of the level curve for the real part of $η(s)=0$ with $η(s)=π^{-s/2}Γ(s/2)ζ^\prime(s)$ gives a new classification of the zeros of $ζ(s)$ and of $ζ^\prime(s)$. We conjecture that for type 2 zeros, $\liminf (β^\prime -1/2)\logγ^\prime = 0$ if and only if $\liminf (γ^+-γ^-)\log γ^\prime=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of $ζ^\prime(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

Level curves for Zhang's Eta Function

TL;DR

This work introduces a level-curve framework to classify zeros of the Riemann zeta function and its derivative via η(s)=π^{−s/2}Γ(s/2)ζ′(s). Under the Riemann Hypothesis with simple zeros, zeros of ζ′(s) are partitioned into three types (0,1,2) based on how the Re(η)=0 level curves exit toward the critical line, and a canonical mapping ties these to zeros of ζ(s), with N2(T)=N2′(T) and N1(T)=N1′(T). The paper provides asymptotic count relations, extensive computational data for ~1e6 zeros near T=10^{10}, and curvature-based arguments that connect liminf behaviors of (β′−1/2)log γ′ and (γ^+−γ^−)log γ′ for type-2 zeros, reducing a conjecture to a curvature bound on the level curve. An explicit Marden-type decomposition and a curvature identity show how the positions of all auxiliary zeros influence the curvature at each type-2 zero. An appendix extends the approach to random matrix analogs, drawing parallel results for zeros of characteristic polynomials of unitary matrices and their derivatives, reinforcing the deep structural similarity between zeta- and matrix-analytic spectra.

Abstract

Study of the level curve for the real part of with gives a new classification of the zeros of and of . We conjecture that for type 2 zeros, if and only if , and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify zeros of near . The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

Paper Structure

This paper contains 10 sections, 17 theorems, 122 equations, 7 figures.

Table of Contents

  1. Classification of zeros
  2. Data
  3. Two Lemmas
  4. A closer look at the type 2 zeros. Curvature of the level curve.
  5. Zeros of $\zeta^{\prime\prime}(s)$
  6. Adaptation of a Theorem of Marden
  7. Curvature at $\rho^\prime$ in terms of all $\lambda^\prime\ne\rho^\prime$
  8. Appendix: Random Matrix Analogs
  9. Results about location of zeros $p_A(z)$ and $p_A^\prime(z)$
  10. Classification of zeros of $p_A(z)$ and $p_A^\prime(z)$.

Key Result

Lemma 1

(Zhang) Suppose $t>7$. Then we have $\zeta(1/2+it)=0$ if and only if $\text{Re}(\eta(1/2+i t))=0$.

Figures (7)

  • Figure 1: Level curves for $\text{Re}(\eta(s))=0$, $-7\le\sigma\le 8$, $0\le t\le 240$
  • Figure 2: $(\beta^\prime-1/2)\log(\gamma^\prime)$ for $10^6$ zeros near $T=10^{10}$.
  • Figure 3: Normalized gaps between pairs of type 2 zeros of $\zeta(s)$.
  • Figure 4: Histogram of $\beta^\prime$ (unscaled) for type 0 zeros.
  • Figure 5: Position of $\rho^{\prime\prime}$ relative to $\rho^\prime$, shown by quartiles of $(\beta^\prime-1/2)\log \gamma^\prime$
  • ...and 2 more figures

Theorems & Definitions (33)

  • Lemma 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Conjecture
  • Conjecture
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 23 more