Density estimates

Every non-trivial zero of the Riemann zeta function ζ(s) has real part 1/2. Equivalent forms involve the ξ-function, the explicit formula, and zero-density estimates on the critical strip.

GATE gt_rh_density: the classical Ingham–Huxley bound N(σ,T)≪T^{c(1−σ)^{3/2}}(log T)^A is now stated and proposed as the —gate→ Density estimates

GATE gt_rh_density: the closed door is addressed by the committed Ingham–Huxley density estimate plus the detection brid —gate→ Density estimates

GATE UPDATE (honest): gt_rh_density now has a clean classical opening lemma — Huxley N(σ,T) ≪ T^{c(1−σ)^{3/2}}(log T)^A —gate→ Density estimates

Cycle 297 RH scaffolding (Gonek 1984 mean values of ζ and derivatives, Trans. AMS 281) imported as PROPOSED classical le —gate→ Density estimates

GATE STATUS (honest): gt_rh_critical remains OPEN. Classical scaffolding now includes ξ/Hadamard, explicit formula, zero —gate→ Density estimates

GATE STATUS (honest): Riemann hypothesis / all non-trivial zeros on Re(s)=1/2 remains OPEN. Classical Hardy (1914) gives —gate→ Density estimates

GATE STATUS (honest): full Riemann Hypothesis / proportion of zeros on Re(s)=1/2 → 1 remains OPEN. Levinson 1974 gives o —gate→ Density estimates

GATE STATUS (honest): Ingham classical zero-density controls N(σ,T) away from the line in a power-saving regime but does —gate→ Density estimates

GATE STATUS (honest): Bohr–Landau 1914 shows almost all nontrivial ζ zeros approach Re=1/2 (N(σ₀,T)=o(N(T)) for σ₀>1/2). —lemma→ Density estimates

GATE STATUS (honest): Vinogradov–Korobov gives a classical zero-free region near Re(s)=1 and strong PNT error terms, but —lemma→ Density estimates

GATE STATUS (honest): Riemann–von Mangoldt gives the classical asymptotic for N(T) in the critical strip but does NOT pr —lemma→ Density estimates

GATE STATUS (honest): Montgomery 1973 pair correlation gives classical / conditional statistical scaffolding for zeros o —lemma→ Density estimates

GATE STATUS (honest): Landau–Page exceptional-zero control near Re(s)=1 does NOT place non-trivial zeros of ζ on Re(s)=1 —lemma→ Density estimates

GATE STATUS (honest): Heath-Brown 1979 zero-density N(σ,T) ≪ T^{12/5(1-σ)}(log T)^A is classical scaffolding and does NO —lemma→ Density estimates

GATE STATUS (honest): Báez-Duarte 2003 gives an equivalent discrete L²-density criterion for RH (strengthening Nyman–Beu —gate→ Density estimates

GATE STATUS (honest): de la Vallée Poussin 1899 gives a classical zero-free region near Re(s)=1 and underpins the Prime —gate→ Density estimates

GATE STATUS (honest): Voronin 1975 universality describes vertical approximation of ζ in 1/2 < σ < 1, a landmark value-d —gate→ Density estimates

GATE STATUS (honest): The Riemann–Siegel formula (Siegel 1932) gives an asymptotic for Z(t) on Re(s)=1/2, enabling effic —gate→ Density estimates

GATE STATUS (honest): Weil 1952 gives the classical explicit formula linking ζ-zeros to primes (and related positivity e —gate→ Density estimates

GATE STATUS (honest): Balazard–Saias–Yor 1999 gives an equivalent integral reformulation of RH, but does NOT prove the R —gate→ Density estimates

GATE STATUS (honest): Hardy 1914 establishes infinitely many zeros of ζ on Re=1/2, but does NOT prove the Riemann Hypoth —gate→ Density estimates

GATE STATUS (honest): Bombieri–Vinogradov 1965 establishes average equidistribution of primes in APs up to Q ≤ x^{1/2}/( —gate→ Density estimates

OPEN gate update (Cycle 122): von Koch 1901 gives classical RH ↔ π(x)=li(x)+O(√x log x) equivalence scaffolding; it does —gate→ Density estimates

OPEN gate update (Cycle 128): Littlewood 1914 gives classical RH-conditional Ω± scaffolding for π(x)−li(x); it does NOT —gate→ Density estimates