Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple

Question

According to a conjecture p.4 $|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$ for $0 < \Delta < \frac12$ and $|t| > 2 \pi +1$. Since $\zeta(\overline{s}) = \overline{\zeta(s)}$, this is equvalent to $|\zeta(s)| > |\zeta(1-s)| $ for $0 < \sigma < \frac12$ and $t$ large enough.

Set $$ F(s) = {\frac {\Gamma \left( 1/2-1/2\,s \right) {\pi }^{-1/2+1/2\,s}{\pi }^{ 1/2\,s}}{\Gamma \left( 1/2\,s \right) }}$$

Then from the functional equation $$\frac{\zeta(s)}{\zeta(1-s)} = F(s) $$ and

$$\frac{|\zeta(s)|}{|\zeta(1-s)|} = |F(s)| $$

$|F(s)| > 1$ for $0 < \sigma < \frac12$ and $t$ large enough would imply the conjecture unless $s$ is a zero of zeta off the critical line.

In Maple 13 using with(MultiSeries); and assuming $0 < \sigma < 1/2$ we get:

$$ \lim_{t \to \infty} |F(\sigma+ it)| = \infty , \, 0 < \sigma < 1/2 $$

Looks like if Maple's result is correct this would mean the conjecture is true at infinity, unless $s$ is a zero off the critical line.

Is Maple's result true?

Proof that $|F(s)| > 1$ for $t$ large enough (except at zeros)?

Answer 1

For fixed $t>12$, let us consider for $0\leq\sigma\leq \frac{1}{2}$ the function

$$ G(\sigma):=|\pi^{-(\sigma-it)/2}\Gamma(\sigma+it)|^2 = \pi^{-\sigma}|\Gamma(\sigma+it)|^2. $$

Following the accepted answer here, we see that

$$ \frac{d}{d\sigma}\log G(\sigma)=-\sigma\log\pi + \psi(\sigma+it) + \psi(\sigma-it)$$

$$\geq -\frac{1}{2}\log\pi + 2(1 - \gamma) - \sum_{n=1}^{\infty} \frac{2}{n^2 + t^2} > 0.27-\frac{\pi}{t}>0,$$

whence $G(\sigma)$ is increasing on $[0,\frac{1}{2}]$.

It follows that $|F(s)|>1$ for $0<\Re(s)<\frac{1}{2}$ and $|\Im s|>24$.

Answer 2

GH has answered your question, but here is some additional remarks.

The function you are calling $F(s)$ is often called $\chi(s)$ in the theory of the Riemann zeta-function (e.g. in Titchmarsh's book The Theory of the Riemann Zeta-Function). Standard asymptotic estimates for the gamma function imply that $$ \chi(s) = \Big( \frac{2\pi}{t} \Big)^{\sigma+it-1/2} e^{i(t+\pi/4)} \Big\{ 1 + O\Big(\frac{1}{t}\Big) \Big\} $$ in any fixed strip $\alpha \le \sigma \le \beta$ as $t \to \infty$ (for $s=\sigma+it$), which also answers your question -- though not as precisely as in GH's answer.

Answer 3

In fact, |F(s)|>1 is more than a conjecture, and it is always satisfied, regardless of whether or not there exist non-trivial zeros off the critical line (even in case there exist some, those would result in removable singularities). In this arxiv.org/pdf/0907.2426v5.pdf I have been studying a ratio strictly related to the inverse of your F(s). Said ratio is taken between the corresponding Dirichlet eta functions. It is graphically represented in Fig. 2, which also shows the upper and lower bound "sheets".