# Is this equivalent to the Riemann Hypothesis?

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 ?

• Increasing/decreasing along what path? – Dzoooks Dec 24 '18 at 20:40
• @Dzoooks, i had forgotten to add certain details, please see the present form of the question. – OneTwoOne Dec 24 '18 at 22:11
• Riemann's functional equation gives a simple relation between $\zeta(s)$ and $\zeta(1-s)$ namely $\frac{\zeta(s)}{\zeta(1-s)} = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)$ – Winther Dec 24 '18 at 22:46
• 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. – Winther Dec 24 '18 at 22:53
• @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 $\zeta(1-s)$. – OneTwoOne Dec 24 '18 at 22:55