A Reformulation of the Riemann Hypothesis
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866915259902590976 |
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| author | Sousa, Jose Risomar |
| author_facet | Sousa, Jose Risomar |
| contents | We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to the right half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $φ(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_08558 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A Reformulation of the Riemann Hypothesis Sousa, Jose Risomar Number Theory 11-XX We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to the right half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $φ(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function. |
| title | A Reformulation of the Riemann Hypothesis |
| topic | Number Theory 11-XX |
| url | https://arxiv.org/abs/2012.08558 |