Cfrgtkky

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he took in deriving and proving his inequality. A similar approach was also taken and expanded upon by F. Saidak and P. D. Zvengrowski in their paper, "On the modulus of the Riemann Zeta Function in the critical strip."

They essentially prove that for $s=sigma + it$, with $t geq 2pi + 1$, and $frac{1}{2} leq sigma leq 1$,

$$
left|zeta(1-s) right| geq left| zeta(s)right| tag{1}
$$

In this proof they all start with the functional equation for the Riemann Zeta Function, say:
$$
zeta(1-s)=(2pi)^{-2s}cosleft(frac{pi s}{2}right)Gamma(s)zeta(s) tag{2}
$$
And go on to show that

$$
left|(2pi)^{-2s}cosleft(frac{pi s}{2}right)Gamma(s)right| geq 1 tag{3}
$$

by looking at how the elementary functions $left|(2pi)^{-2s}right|$ and $left|cosleft(frac{pi s}{2}right)right|$ change with $sigma$ compared to $left|Gamma(s)right|$. The elementary functions decrease as $sigma$ increases, " leaving the whole burden of increase upon the $Gamma$-function," as Spira puts it. The next bit confuses me however, as they say they require,

$$
frac{partial}{partialsigma}left( left|(2pi)^{-2s}cosleft(frac{pi s}{2}right)Gamma(s)right|right) gt 0 tag{4}
$$

Where differentiating the elementary functions is somewhat trivial compared to the $Gamma$-function, which requires use of the Stirling Approximation, which makes things a lot more complicated and introduces an error factor. Therefore, my question is whether for inequality $(3)$, it is simpler and valid to firstly rearrange for the $Gamma$-function:

$$
left|Gamma(s)right| geq frac{1}{left|(2pi)^{-2s}cosleft(frac{pi s}{2}right)right|}
$$

Square it,

$$
left|Gamma(s)right|^{2} geq frac{1}{left|(2pi)^{-2s}cosleft(frac{pi s}{2}right)right|^{2}}
$$

Which can be simplified to:

$$
left|Gamma(s)right|^{2} geq frac{left(2piright)^{2sigma}}{2 left( cosh(pi t) + cos(pi sigma)right)} tag{5}
$$

Now clearly for $sigma = frac{1}{2}$ this becomes:

$$
left|Gammaleft(frac{1}{2}+itright)right|^{2} = frac{pi}{cosh(pi t)} tag{6}
$$

And so the proof boils down to showing that for $frac{1}{2} lt sigma leq 1$, that:

$$
left|Gamma(s)right|^{2} gt frac{left(2piright)^{2sigma}}{2 left( cosh(pi t) + cos(pi sigma)right)} tag{7}
$$

Which excludes the need for differentiation as well as the more complicated Stirling Approximation, and the error factor that comes with it, that inevitably limits the proof by "always exceed[ing] the margin of safety of the inequality," as Spira says.

The reason I ask if this is a valid approach, is because then $left| Gamma(s) right|^{2}$ is a much nicer term to work with due to the functional equation for the $Gamma$-function:

$$
Gamma(1-s)Gamma(s) = frac{pi}{sinh(pi s)}
$$

As from it we have equation $(6)$, but also:

$$
left|Gammaleft(1+itright)right|^{2} = frac{pi t}{sinh(pi t)}
$$

It is also known that $left| Gamma(s) right|^{2} leq left| Gamma(sigma) right|^{2}$ where the equality is for $t=0$. This combined with the fact that the minimum of $left|Gamma(sigma)right|^{2}$ occurs at $sigma gt 1$ suggests that the inequality $(7)$ can be taken to its extreme at $sigma = 1$ with $left| Gamma(s) right|^{2}$ being at its minimum point for $frac{1}{2} lt sigma leq 1$ and the RHS being at its maximum, i.e $(7)$ would become:

$$
frac{pi t}{sinh(pi t)} gt frac{4 pi^{2}}{2 left( cosh(pi t) - 1right)}
$$

Simplifying the equation and we get:

$$
t gt 2 pi cothleft(frac{pi t}{2}right)
$$

where $coth(pi t)$ quickly limits to $1$ as $t gt 2$, and so we get:

$$
t gt 2 pi
$$

For which the inequality $(7)$ holds, presumably excluding any kind of ambiguity. Would this then mean that the inequality in $(1)$ can be strengthened to exclude the equality over the range $frac{1}{2} lt sigma leq 1$ and be only equal for $sigma = frac{1}{2}$ for $t gt 2pi$?

Now I have a gut feeling that something (or a few things) above is incorrect and that I have misinterpreted a result in the mentioned papers and/or have done something incorrect in my maths, either explicitly or in the form of an incorrect assumption made. So, I would kindly ask for some clarification. Thanks!

Edit 06/12/18: My reasoning that the LHS of $(7)$ is at its minimum at $sigma = 1$ is incorrect as for $t gt gt 1$, $left| Gamma(s) right|^{2}$ is smaller for $sigma < 1$. However, here I found an alternative approximation for $left| Gamma(s) right|^{2}$ and found that the error in the approximation approaches zero as $t rightarrow infty$ and is valid for $t gt 2pi + 1$. Of course, this is assuming the maths is correct.