A Reformulation of the Riemann Hypothesis

Jose Risomar Sousa

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arXiv · 2012.08558

A Reformulation of the Riemann Hypothesis

Jose Risomar Sousa

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Abstract

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, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.

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Jose Risomar Sousa. 2020-12-15. A Reformulation of the Riemann Hypothesis. https://arxiv.org/abs/2012.08558

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