  We present some novelties on the Riemann zeta function. Using the analytic
continuation we created for the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, we
extend the zeta function from $\Re(k)>1$ to the complex 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.
