Does normal Riemann hypothesis for $\zeta_{\mathbb{Q}}$ follows from the extended Riemann hypothesis for some $K \neq \mathbb{Q}$ (i.e. the statement that all zeroes of the Dedekind zeta function $\zeta_K$ for $K$ a number field in the critical strip lie on the axis $\mathfrak{R}(s)=1/2$)?
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$\begingroup$ Well, $\mathbb{Q}$ is a number field... $\endgroup$– Qiaochu YuanCommented Aug 18, 2012 at 0:12
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1$\begingroup$ Yes I mean if we know it is true for one $K \neq \mathbb{Q}$ does it follow for $\mathbb{Q}$? $\endgroup$– user16974Commented Aug 18, 2012 at 0:13
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Yes, certainly: zeta functions of abelian extensions of $\mathbb Q$ factor as products of Dirichlet $L$-functions over $\mathbb Q$, including $\zeta(s)$, and those other $L$-functions have no poles (which might cancel an off-line zero of zeta), so RH for any abelian extension of $\mathbb Q$ implies that for $\mathbb Q$.
At this point in history, the fact that we do not know that all Artin $L$-functions lack poles in the critical strip severely complicates our provable understanding of an analogous assertion for not-abelian extensions.
answered Aug 18, 2012 at 0:26
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$\begingroup$ Nonabelian Galois extensions still work. That is, if $L/K$ is a finite Galois extension of number fields, then $\zeta_L(s)/\zeta_K(s)$ is known to be analytic. $\endgroup$ Commented Jun 24, 2014 at 10:05