By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|\zeta(1-s)|$ increases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ fixed.
Since $|\zeta(s)|$ is continuous, decreasing for $\Re(s)>1$ and $s$ is a zero of $\zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|\zeta(s)|$ decreases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ fixed.
A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=\frac{|\zeta(s)|}{|\zeta(1-s)|}$ decreases as $\Re(s)$ varies on $(1/2, 1]$ with $|t|=|\Im(s)|\geq 165$ fixed ?