I am working from some notes concerning bounds on Dedekind zeta functions and am trying to derive a supposed version of Stirling's approximation contained therein:
Let $\sigma$ be fixed and $|t|\rightarrow \infty$, then a version of Stirling's approximation formula yields
$$|\Gamma(\sigma+it)| \sim |t|^{\sigma-\tfrac{1}{2}}e^{-\tfrac{\pi}{2}|t|}\ll |t|^{\sigma-\tfrac{1}{2}}e^{-|t|}$$
where we define
$$f(z)\ll g(z) \text{ iff } \exists c\in \mathbb{R}: \exists z_0\in \mathbb{C}: \forall |z|>|z_0|: |f(z)|<c|g(z)|$$
I started working from what appears to be the standard rendering of Stirling:
$$\Gamma(s)=\frac{\sqrt{2\pi}s^{s-\frac{1}{2}}}{e^s}\big(1+O(\tfrac{1}{s})\big) \qquad \underset{\text{Since }O\big(\tfrac{1}{s}\big)\ll \tfrac{1}{s} \text{ by definition}}{\Rightarrow} \qquad \Gamma(s)\ll \Big|\frac{\sqrt{2\pi}s^{s-\frac{1}{2}}}{e^s}\Big|\cdot (1+|\tfrac{1}{s}|)$$
But making the relevant substitutions, shuffling things around, and ignoring the negligible contribution of $\sigma$ to the modulus when applicable, I have only been able to find
$$|\Gamma(\sigma+it)|\ll |t|^{\sigma-\tfrac{1}{2}}+|t|^{\sigma-\tfrac{3}{2}}$$
Best Answer
The decay comes from $z^z=\exp(z \log(z))$. When $|t/\sigma|$ is large, $\log(z)$ has imaginary part close to $\operatorname{sign}(t) \pi/2$, so that the real part of the exponent behaves like $\sigma \log(\sigma) - \pi/2 |t|$ which is dominated by the second term, giving the decay.