A question about Speiser's 1934 result on the Riemann hypothesis - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:51:13Z http://mathoverflow.net/feeds/question/107938 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107938/a-question-about-speisers-1934-result-on-the-riemann-hypothesis A question about Speiser's 1934 result on the Riemann hypothesis pbs 2012-09-23T21:43:57Z 2012-09-30T08:46:53Z <p>A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0&lt;\text{Re}(s)&lt;1/2$. But I have seen some (possibly less reliable) sources without proof suggesting this is an <em>if and only if</em> 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 <em>if and only if</em> relationship or not?</p> http://mathoverflow.net/questions/107938/a-question-about-speisers-1934-result-on-the-riemann-hypothesis/107941#107941 Answer by Mark Lewko for A question about Speiser's 1934 result on the Riemann hypothesis Mark Lewko 2012-09-23T22:02:59Z 2012-09-23T22:02:59Z <p>Yes, Speiser's theorem is an if and only if.</p> <p>See Theorem 1 and "Corollary to Theorem 1" in Levinson and Montgomery's <a href="http://www.springerlink.com/content/dk48528205t15585/fulltext.pdf" rel="nofollow">Zeros of the derivatives of the Riemann Zeta-function</a>. Acta Math. 133 (1974), 49–65. </p>