An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function
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arXiv
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| Format: | Preprint |
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2023
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| author | Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. |
| author_facet | Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. |
| contents | Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros $ρ=β+iγ$ of the Riemann zeta-function such that $T^{3/8}<γ\le T$ satisfy $|β-1/2|<1/(2\log T)$, %lie in the thin box $\{s=σ+it: |σ-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$, then, as $T$ tends to infinity, at least $61.7\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where $β=1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04799 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function Baluyot, Siegfred Alan C. Goldston, Daniel Alan Suriajaya, Ade Irma Turnage-Butterbaugh, Caroline L. Number Theory Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple. Here we obtain an unconditional form of Montgomery's theorem and show how to apply it to prove the following result on simple zeros: Assuming all the zeros $ρ=β+iγ$ of the Riemann zeta-function such that $T^{3/8}<γ\le T$ satisfy $|β-1/2|<1/(2\log T)$, %lie in the thin box $\{s=σ+it: |σ-1/2|<1/(2\log T),\ T^{3/8}<t\le T\}$, then, as $T$ tends to infinity, at least $61.7\%$ of these zeros are simple. The method of proof neither requires nor provides any information on whether any of these zeros are on or not on the critical line where $β=1/2$. We also obtain the same result under the weaker assumption of a strong zero-density hypothesis. |
| title | An unconditional Montgomery Theorem for Pair Correlation of Zeros of the Riemann Zeta Function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.04799 |