grand_riemann_hypothesis

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

The Grand Riemann Hypothesis: All non-trivial zeros of Dirichlet L-functions L(s, χ) have real part 1/2. The Riemann Hypothesis is the case of the trivial character. Known for 1-dim characters; the generalization to all automorphic L-functions is the most comprehensive form.

Why this node was retired

The posted statement is

import Mathlib

theorem grand_riemann_hypothesis (chi : ZMod 5 → ℂ)
    (hchar : ∀ a b : ZMod 5, chi (a * b) = chi a * chi b)
    (hne0 : chi 1 ≠ 0) :
    ∀ s : ℂ, (∑' n : ℕ, chi n / (n + 1 : ℂ) ^ s = 0) →
      s.re = 1 / 2 := by
  sorry

An unrestricted raw series is substituted for an analytically continued L-function, with no critical-strip restriction.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

The existing formula only suggests a Dirichlet-L version: use genuine Dirichlet characters and analytically continued L-functions, and restrict to nontrivial zeros in 0<Re(s)<1. A general automorphic or Artin-L GRH statement needs its own specified class of L-functions and cannot be inferred from a function on ZMod 5.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.