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A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) sources without proof suggesting this is an if and only if relationship, i.e. RH$\Longleftrightarrow\zeta'(s)\neq 0$. However, those (perhaps more reliable) sources state only forward implication, i.e. RH$\Longrightarrow\zeta'(s)\neq 0$. My question is this: is Speiser's result an if and only if relationship or not?

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Yes, Speiser's theorem is an if and only if.

See Theorem 1 and "Corollary to Theorem 1" in Levinson and Montgomery's Zeros of the derivatives of the Riemann Zeta-function. Acta Math. 133 (1974), 49–65.

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