Improved zero-density estimates for the Riemann zeta function
We establish improved zero-density estimates for the Riemann zeta function relative to the bounds previously released in PALOMAR-2026-10-11-000010 v1 and PALOMAR-2026-10-09-000005 v1. Let N(σ,T) count the nontrivial zeros with Re ρ ≥ σ and |Im ρ| ≤ T, with analytic multiplicity. For T ≥ 2 and ε > 0, our bounds have the form N(σ,T) ≤ C(σ,ε)T^((1−σ)A(σ)+ε). The native quantitative bounds improve those earlier power curves for 12/17 < σ ≤ 3/4 and 7/9 < σ ≤ 7/8. The displayed power-curve comparison continues for 7/8 < σ < 17/19, where the imported OpenAI result already gives N(σ,T)=0. Later…
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