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Martin Sleziak
Martin Sleziak
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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$\Leftrightarrow\zeta'(s)\neq 0$$\Longleftrightarrow\zeta'(s)\neq 0$. However, those (perhaps more reliable) sources state only forward implication, i.e. RH$\Rightarrow\zeta'(s)\neq 0$$\Longrightarrow\zeta'(s)\neq 0$. My question is this: is Speiser's result an if and only if relationship or not?

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$\Leftrightarrow\zeta'(s)\neq 0$. However, those (perhaps more reliable) sources state only forward implication, i.e. RH$\Rightarrow\zeta'(s)\neq 0$. My question is this: is Speiser's result an if and only if relationship or not?

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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A question about Speiser's 1934 result on the Riemann hypothesis

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$\Leftrightarrow\zeta'(s)\neq 0$. However, those (perhaps more reliable) sources state only forward implication, i.e. RH$\Rightarrow\zeta'(s)\neq 0$. My question is this: is Speiser's result an if and only if relationship or not?