A Reformulation of the Riemann Hypothesis

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, ${\mathrm{L}\mathrm{i}}_k(e^z)$, the zeta function’s Dirichlet series is analytically continued from $\mathrm{\Re }(k)>1$ to the right half-plane, $\mathrm{\Re }(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta’s cousin, $\phi (k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.