Abstract:
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.
