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

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 $7/8$).

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 $7/8$).