Is it true that $\left|\frac{\zeta(0.5+it) }{\zeta(0.5-it)}\right|\leq 1$, for $t\in \mathbb{R}$?

I'm interested one Bounds values of Riemann zeta function on critical line , really i have got this from some computation I did in wolfram alpha for some values of $$t$$ and according to the studying number of solution of : $$\zeta(0.5+it )= z$$ , for every real $$t$$ and $$z \in \mathbb{C}$$ this Bounds :

$$\left|\frac{\zeta(0.5+it) }{\zeta(0.5-it)}\right|\leq 1$$

Now if we assume RH Holds the inequality is become an equality to $$1$$ by passage of limit and it is true if $$(t=0)$$ , But what about $$t \neq 0$$ ?

• Isn't it always equal to $1$? Aug 29, 2019 at 17:08
• @Lord Shark the Unknown is right. No matter if the RH holds, it is always equal to 1 Aug 29, 2019 at 17:13
• If it was not equal to one, then the inequality could not hold. Just replace $t$ by $-t$. Aug 29, 2019 at 19:39
• $\zeta(1/2+it)$ and $\zeta(1/2-it)$ have the same modulus since they are complex conjugates. Aug 29, 2019 at 20:45

Any meromorphic function $$f$$ real-valued on the real line has the property that $$f(\overline s)=\overline{f(s)}$$, so the absolute values are identical. It's nothing special to zeta.