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By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|\zeta(1-s)|$ increases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ fixed.

Since $|\zeta(s)|$ is continuous, decreasing for $\Re(s)>1$ and $s$ is a zero of $\zeta$ whenever $(1-s)$ is a zero, Spira's result entails that the RH is equivalent to the statement that $|\zeta(s)|$ decreases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ fixed.

A combination of these two statements seems to yield another equivalent statement for the RH, namely: *The RH is equivalent to the statement that $F(s)=\frac{|\zeta(s)|}{|\zeta(1-s)|}$ decreases as $\Re(s)$ varies on $(1/2, 1]$ with $|t|=|\Im(s)|\geq 165$ fixed ?

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  • $\begingroup$ Increasing/decreasing along what path? $\endgroup$ – Dzoooks Dec 24 '18 at 20:40
  • $\begingroup$ @Dzoooks, i had forgotten to add certain details, please see the present form of the question. $\endgroup$ – OneTwoOne Dec 24 '18 at 22:11
  • $\begingroup$ Riemann's functional equation gives a simple relation between $\zeta(s)$ and $\zeta(1-s)$ namely $\frac{\zeta(s)}{\zeta(1-s)} = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)$ $\endgroup$ – Winther Dec 24 '18 at 22:46
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    $\begingroup$ btw it's a one way implication if $|\zeta(s)|$ is decreasing and $|\zeta(1-s)|$ is increasing $\implies $ $F(s)$ is decreasing. You don't get the other way so it's not an equivalent formulation. $\endgroup$ – Winther Dec 24 '18 at 22:53
  • $\begingroup$ @Winther, basing on the fact that Spira didn't use any other property of $\zeta$ besides the functional equation, i wouldn't perceive the functional equation as a ''simple'' relation between $\zeta(s)$ and $\zeta(1-s)$. $\endgroup$ – OneTwoOne Dec 24 '18 at 22:55

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