A quasi-Riemann hypothesis for Dedekind zeta functions
Recent work on Artin's primitive root conjecture establishes a zero-free half-plane $\mathrm{Re}\,s>1-10^{-6}$ for finite-order Hecke $L$-functions over cyclotomic fields containing $e^{2\pi i/12}$. Following their strategy, we prove that if $K$ is any number field, then the Dedekind zeta function $\zeta_K(s)$ does not vanish in the half-plane $\mathrm{Re}\,s>31/32$. As a corollary, the $L$-function of any finite-order Hecke character defined over $K$ also does not vanish in the half-plane $\mathrm{Re}\,s > 31/32$, wherever it is holomorphic. These results were proved and formalized in Lean…
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