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Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, Caroline L. Turnage-Butterbaugh

TL;DR

This work extends Montgomery's pair-correlation framework beyond the Riemann Hypothesis by analyzing horizontal distribution of zeta zeros under a thin vertical box hypothesis: all zeros with $T<\gamma\le 2T$ lie in $B_b$ of width $\tfrac{b}{\log T}$ around the critical line. By combining diagonal and symmetric-diagonal contributions with carefully chosen kernels, the authors establish that at least $\tfrac{2}{3}$ of zeros are simple and at least $\tfrac{2}{3}$ lie on the critical line, and they prove at least $\tfrac{1}{3}$ of zeros are both simple and on the line; these results hold under the box hypothesis and yield explicit constants for small $b$ values. The Tsang kernel further sharpens the bounds, showing that at least $67.25\%$ of zeros lie on the critical line and at least $34.5\%$ are both simple and on the line under the same hypothesis, while an unconditional Montgomery framework remains available for bounding zero-distributions. Overall, the paper demonstrates the versatility of pair-correlation methods to extract horizontal distribution information about zeta zeros and provides concrete, numerically explicit bounds via kernel techniques.

Abstract

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least $2/3$ of the zeros are simple. In this paper, we investigate the versatility of the pair correlation method and show, for the first time, that it can be used to prove results on the \emph{horizontal distribution} of zeros of the Riemann zeta-function. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros $ρ= β+iγ$ with $T<γ\le 2T$ are in a narrow vertical box centered on the critical line with width $b/\log T$, where $b\to 0$ as $T\to \infty$. We first prove the generalization of Montgomery's result that at least $2/3$ of zeros are simple, and we then prove the new result that the pair correlation method yields at least $2/3$ of the zeros on the critical line. We also use the pair correlation method to prove that at least $1/3$ of the zeros are both simple and on the critical line, a result already known unconditionally using different methods.

Pair Correlation of Zeros of the Riemann Zeta Function I: Proportions of Simple Zeros and Critical Zeros

TL;DR

This work extends Montgomery's pair-correlation framework beyond the Riemann Hypothesis by analyzing horizontal distribution of zeta zeros under a thin vertical box hypothesis: all zeros with lie in of width around the critical line. By combining diagonal and symmetric-diagonal contributions with carefully chosen kernels, the authors establish that at least of zeros are simple and at least lie on the critical line, and they prove at least of zeros are both simple and on the line; these results hold under the box hypothesis and yield explicit constants for small values. The Tsang kernel further sharpens the bounds, showing that at least of zeros lie on the critical line and at least are both simple and on the line under the same hypothesis, while an unconditional Montgomery framework remains available for bounding zero-distributions. Overall, the paper demonstrates the versatility of pair-correlation methods to extract horizontal distribution information about zeta zeros and provides concrete, numerically explicit bounds via kernel techniques.

Abstract

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least of the zeros are simple. In this paper, we investigate the versatility of the pair correlation method and show, for the first time, that it can be used to prove results on the \emph{horizontal distribution} of zeros of the Riemann zeta-function. In earlier work we showed how to remove RH from Montgomery's theorem and, in turn, obtain results on simple zeros assuming conditions on the zeros that are weaker than RH. Here we assume a more general condition, namely that all the zeros with are in a narrow vertical box centered on the critical line with width , where as . We first prove the generalization of Montgomery's result that at least of zeros are simple, and we then prove the new result that the pair correlation method yields at least of the zeros on the critical line. We also use the pair correlation method to prove that at least of the zeros are both simple and on the critical line, a result already known unconditionally using different methods.
Paper Structure (10 sections, 6 theorems, 82 equations, 2 tables)

This paper contains 10 sections, 6 theorems, 82 equations, 2 tables.

Table of Contents

  1. Introduction and Statement of the Main Results
  2. Montgomery's method and critical zeros
  3. Sketch of the approach
  4. Statements of results
  5. Unconditional Montgomery Theorem
  6. Proof of \ref{['MT']}
  7. Evaluation of a Sum over differences of zeros by Montgomery's Theorem
  8. The Tsang Kernel
  9. Proof of Theorem \ref{['thm1']}
  10. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Assume that, for all sufficiently large $T$, all the zeros $\rho = \beta +i\gamma$ of $\zeta(s)$ with $T<\gamma \le 2T$ are in $B_b$. Then we have, where $b\to 0$ as $T\to \infty$, and

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • proof : Proof of Theorem 1 in BGST-PC
  • Lemma 1
  • proof : Proof of \ref{['Mon-1']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem2']}
  • Remark 3
  • ...and 5 more