Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$

Question

When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $ Re(s)=\frac12 $ and $t >0$ I come up to the final Bounds for $\zeta(0.5+it) $ which is represented by the following formual :For $t\geq 1.22$: $$|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}\tag{1}$$, For Bounds of $\Gamma(s)$ it is found that is monotonic increasing function for $|t|\geq 5/4$ with the respect to the the real part of $s$ and it were false with $|t|\leq 1$ in this paper entitled On the Horizontal Monotonicity of $|\Gamma(s)|$ by Gopala Krishna Srinivasan and P. Zvengrowski, |$\Gamma(s)$| is given in the introduction of that paper for $s=\sigma+ i t$ by this formula : $|\Gamma(\sigma+ i t)|=\lambda \frac{\Gamma(1+\sigma)}{\sqrt{\sigma^2+t^2}}\sqrt{\frac{2\pi t}{\exp(\pi t)-\exp(-\pi t)}},\lambda \in(1,1+\sqrt{1+t^2})\tag{2}$, it seem the Right hand side of that formual related to cos hyperbolic function , Now When I tried to plug this formual in the RHS of my bounds it give me a complicated form such that no simple formula for simplification , My question here How I can simplify RHS OF $1$ if it is true ?

Note: The motivation of this question is to look for some connections of primes distribution to Gaussian distribution.

Answer 1

The RHS of (1) is $\sim |t|$ by Stirling's formula. The Weyl bound states that there exists a constant $c>0$ (which one can compute, but I won't) such that if $t\in\mathbb{R}$, then $|\zeta(\frac{1}{2}+it)|\leq c(|t|+1)^{1/6}$. One can do better under assuming the Riemann hypothesis, see Chandee and Soundararajan.