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For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $S_n(s)$ the $n^{th}$partial sum of the alternating Dirichlet series $1-2^{-s}+3^{-s}-... $ . Wefirst show that $S_n(s)\neq 0$ for all $n$ greater than some index $N(s)$ .Denoting by $D={s \in \mathbb{C}:  0< \Re(s) < {1/2}}$ the open left half of the critical strip, define for all$s\in D$ and $n>N(s)$ the ratio $P_n(s) = S_n(1-s) / S_n(s)$ . We then provethat the limit $L(s)=\lim_{N(s)<n\to\infty}P_n(s)$ exists at every point $s$ ofthe domain $D$ . Finally, we show that the function $L(s)$ is continuous on $D$if and only if the Riemann Hypothesis is true.

A Reformulation of the Riemann Hypothesis in Terms of Continuity of the  Limit Function of a Certain Ratio of Partial Sums of a Series for the  Dirichlet Eta Function