Monotonicity of the Riemann zeta function and related functions

We shall consider monotonicity in the “horizontal direction” for several well known functions $f(s)$ of the complex variable $s = \sigma + it$, where monotonicity here means $|f(s)|$ is monotone increasing or monotone decreasing as $\sigma$ increases. The first function will be the well known gamma function $\Gamma(s)$, and it will be shown that $|\Gamma(s)|$ is monotone increasing in $\sigma$ once one is a small distance away from the real axis, more precisely for $|t| > 5/4$. A similar result will be shown for the Riemann zeta function $\zeta(s)$ as well as the two related functions $\eta(s)$ (the Euler–Dedekind eta function) and $\xi(s)$ (the Riemann $\xi$ function). Here it will be shown that all three have monotone decreasing modulus for $\sigma < 0$ and $|t| > 8$, and that for any of the three functions the extension of this monotonicity result to $\sigma < 1/2$ is equivalent to the Riemann Hypothesis. An inequality relating the monotonicity of all three functions will be given.