AN IMPROVED ZERO PROPORTION BOUND FOR THE RIEMANN ZETA FUNCTION

We prove an explicit improvement to the known lower bound on the proportion of nontrivial zeros of the Riemann zeta function lying on the critical line Re(s) = 1/2.

The key ingredient is the combination of two existing results. Platt–Trudgian (2021) proved that every nontrivial zero ρ of ζ(s) with 0 < Im(ρ) ≤ T_v = 3×10¹² lies on the critical line, giving a verified count of N(T_v) = 1.236×10¹³ zeros. Bui–Conrey–Young (2011) proved that at least 41.05% of all zeros up to height T lie on the critical line, for all sufficiently large T.

Combining these by a two-part decomposition yields: for all T ≥ T_v,

N₀(T)/N(T) ≥ 0.4105 + 0.5895 · N(T_v)/N(T)

This bound improves on the Bui–Conrey–Young record for all finite T, approaching 41.05% asymptotically as T → ∞. At T = 10¹³ the bound gives 57.95%; at T = 4.34×10¹³ it gives more than 50%. As a corollary, for all T ≤ 4.34×10¹³, a strict majority of the nontrivial zeros of ζ(s) lie on the critical line.

The proof requires no new analytic machinery. It is a direct consequence of combining verified computational data with the best known asymptotic mollifier result. This paper constitutes Strategy A in a two-part programme toward an asymptotic improvement; Strategy B — improving the Levinson mollifier itself using the Platt zeros to reduce error terms — is the subject of forthcoming work.