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Zeta Zeros in a Narrow Vertical Box

Daniel A. Goldston, Ade Irma Suriajaya

Abstract

In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height $T$ and $2T$ of width $b/\log T$ and centered on the critical line, then, if $b=b(T)\to 0$ as $T\to \infty$, we have asymptotically at least 2/3 of the zeros are simple and on the critical line.

Zeta Zeros in a Narrow Vertical Box

Abstract

In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height and of width and centered on the critical line, then, if as , we have asymptotically at least 2/3 of the zeros are simple and on the critical line.

Paper Structure

This paper contains 8 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Convention for counting zeros with multiplicity
  3. Counting zeros in the critical strip
  4. Montgomery's Simple Zero Method without RH
  5. Unconditional Bound for Close Pairs of Zeros
  6. Montgomery's Theorem with and without RH
  7. Unconditional Fejér Kernel Sum over Zero Differences without RH
  8. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Assuming all of the zeros $\rho = \beta + i\gamma$ of $\zeta(s)$ with $T<\gamma \le 2T$ lie in the region Then for the zeros of $\zeta(s)$ with $T<\gamma \le 2T$, we have asymptotically that at least 2/3 are simple and on the critical line.

Theorems & Definitions (8)

  • Theorem 1: Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh
  • Theorem 2
  • Lemma 1
  • Lemma 2: Aryan
  • proof : Proof of Lemma \ref{['lem1']}
  • Lemma 3: Selberg
  • proof
  • proof : Proof of Theorem \ref{['thm1']}