The Landau-Siegel zero problem: a uniform logarithmic gap for real zeros of real Dirichlet L-functions

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Vinogradov's least quadratic nonresidue conjecture evidence for this event

Companion: alternate proof of the zero-free half-plane Re s > 11/12 evidence for this event

Claims an absolute constant $c>0$ such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$-function of conductor $q\ge3$ satisfies $(1-\beta)\log q\ge c$. The proof combines a prime-bias estimate forced by a zero close to $1$ with an interpolation determinant on a biquadratic field $\mathbb Q(\sqrt d,\sqrt2)$. The constant is not made explicit, so the result does not by itself give effective class-number bounds with numbers attached. It excludes real zeros in $1-c/\log q<\beta<1$ only; zeros elsewhere in $(0,1)$ are not addressed by this manuscript (the family's 7/8 manuscript excludes those above $7/8$). parent of this event

William Banks' exposition and simpler proof evidence for this event

Lean: release scope note (states the half-plane, not this corollary) evidence for this event

Wikipedia page about Siegel zeroes evidence for this event

Bryce Kerr's simpler proof of the result evidence for this event

Companion: Uniform exclusion of Landau-Siegel zeros evidence for this event

Terence Tao's exposition of the proof evidence for this event

Lean comparator statement: uniform real-zero gap evidence for this event

The Landau-Siegel zero problem: a uniform logarithmic gap for real zeros of real Dirichlet L-functions parent of this event

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Lean: main declaration exists_absolute_real_zero_gap evidence for this event