On the zeros of Riemann's Xi Function
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914907582103552 |
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| author | Raman, Akhila |
| author_facet | Raman, Akhila |
| contents | We consider Riemann's Xi function $ξ(s)$ which is evaluated at $s = \frac{1}{2} + σ+ i ω$, given by $ξ(\frac{1}{2} + σ+ i ω)= E_{pω}(ω)$, where $σ, ω$ are real and compute its inverse Fourier transform given by $E_p(t)$. We study the properties of $E_p(t)$ and a promising new method is presented which could be used to show that the Fourier Transform of $E_p(t)$ given by $E_{pω}(ω) = ξ(\frac{1}{2} + σ+ i ω)$ does not have zeros for finite and real $ω$ when $0 < |σ| < \frac{1}{2}$, corresponding to the critical strip excluding the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09670 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the zeros of Riemann's Xi Function Raman, Akhila General Mathematics We consider Riemann's Xi function $ξ(s)$ which is evaluated at $s = \frac{1}{2} + σ+ i ω$, given by $ξ(\frac{1}{2} + σ+ i ω)= E_{pω}(ω)$, where $σ, ω$ are real and compute its inverse Fourier transform given by $E_p(t)$. We study the properties of $E_p(t)$ and a promising new method is presented which could be used to show that the Fourier Transform of $E_p(t)$ given by $E_{pω}(ω) = ξ(\frac{1}{2} + σ+ i ω)$ does not have zeros for finite and real $ω$ when $0 < |σ| < \frac{1}{2}$, corresponding to the critical strip excluding the critical line. |
| title | On the zeros of Riemann's Xi Function |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2305.09670 |