The quasi-Riemann hypothesis
The following describes the scope of the Lean formalization related to the following accompanying paper(s):
- The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re(s)>7/8$
- Uniform exclusion of Landau–Siegel zeros
Scope
The quasi-Riemann hypothesis asks for a fixed zero-free half-plane $\Re s>\theta$ with $\theta<1$. The formalization gives $\theta=7/8$ for the Riemann zeta function and every Dirichlet $L$-function, uniformly over all positive moduli and all characters. It also establishes the same bound for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$.
The principal-character poles at $s=1$ are excluded in the Dirichlet and Hecke statements. The paper's later applications are not included.
The formalized result gives a uniform logarithmic exclusion region for Landau–Siegel zeros. There is one constant $c>0$ such that every primitive nonprincipal real Dirichlet character of conductor $q\ge3$ and every real zero $0<\beta<1$ of its $L$-function satisfy $1-\beta\ge c/\log q$.
Both character parities are included. No explicit value of $c$ is given. This excludes real zeros in $1-c/\log q<\beta<1$, but does not rule out real zeros elsewhere in $(0,1)$.
Comparator links
| Result | Comparator statement |
|---|---|
| Riemann zeta $7/8$ bound | QuasiRiemannHypothesis.lean |
| Dirichlet $L$-function $7/8$ bound | DirichletSevenEighths.lean |
| Finite-order Hecke $L$-function $7/8$ bound | HeckeSevenEighths.lean |
| Uniform real-zero gap | SiegelZeros.lean |