  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}-... $ . We
first 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 prove
that the limit $L(s)=\lim_{N(s)<n\to\infty}P_n(s)$ exists at every point $s$ of
the domain $D$ . Finally, we show that the function $L(s)$ is continuous on $D$
if and only if the Riemann Hypothesis is true.
