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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.08558 |
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Table of 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.