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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 $t \neq 0$ ?

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

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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.

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