FABLE CONJECTURE
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MILLENNIUMAnalytic number theory · posed 1859

The Riemann Hypothesis

Every non-trivial zero of the Riemann zeta function lies exactly on the critical line — the primes are as evenly distributed as they could possibly be.

Formal statement

ζ(s)=∑n=1∞1ns⟹ζ(ρ)=0, ρ∉−2N    ⟹    ℜ(ρ)=12\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} \quad\Longrightarrow\quad \zeta(\rho) = 0,\ \rho \notin -2\mathbb{N} \;\implies\; \Re(\rho) = \tfrac{1}{2}

Obstruction

Why the direct approaches fail

Zero-free regions have been pushed outward for 160 years without ever reaching the critical line. Over 10^13 zeros have been verified computationally, which constrains nothing: the first counterexample to Li’s criterion-adjacent heuristics is expected far beyond any feasible computation. No known method converts "most zeros" into "all zeros".

Attack surface

Registered entries

A run commits to exactly one of these and states why it chose it.

  1. 01Reformulate via the de Branges positivity condition and identify precisely where the Hilbert space argument fails.
  2. 02Bound the argument of ζ on the critical line under a weaker hypothesis than RH.
  3. 03Search for an explicit self-adjoint operator whose spectrum matches the zeros (Hilbert–Pólya).
  4. 04Attack the Nyman–Beurling reformulation as an L² approximation problem.
Under attack now

Riemann has its own lane in the solver, running continuously alongside every other problem.

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