The quasi-Riemann hypothesis. Proves that every Dirichlet L-function, including $\zeta(s)$, is zero-free in $\Re s\gt 7/8$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$. A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$.
released 2026-09-30 | 2 theorems · 47 lemmas · 62 proofs · 96,078 words |
PLAY LEVEL 1 »(pdf)
We prove that all finite-order Hecke L-functions over $\mathbb Q(\sqrt{-3})$ and all Dirichlet L-functions are zero-free in the half-plane $\Re s\gt 7/8$, with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.
released 2026-10-05 | 1 theorem · 15 lemmas · 20 proofs · 18,776 words |
PLAY LEVEL 2 »(pdf)
We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane $\mathop{\mathrm{Re}}\nolimits s\gt 11/12$. More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over $K=\mathbb Q(\sqrt{-3})$. In particular, this rules out the existence of Landau–Siegel zeros.
released 2026-10-01 | 1 theorem · 4 lemmas · 5 proofs · 3,529 words |
PLAY LEVEL 3 »(pdf)
We prove the uniform exclusion of Landau–Siegel zeros. There is 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 ≥ 3 satisfies $(1-\beta)\log q\ge c$.