{"id":"palomar/PALOMAR-2026-10-11-000010","url":"https://github.com/mrklwk/zeta-density-39-over-17-lean/tree/ee2a8a32be49a4f73c7386e9dffdea24d7e3d8c0/lean","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-11-000010-v1.json","repo":"https://github.com/mrklwk/zeta-density-39-over-17-lean","tags":["math.NT","analytic-number-theory","riemann-zeta","zero-density"],"score":0,"title":"Zero-density estimates from an improved large-values bound","trust":"high","domain":"math.NT","source":"palomar","authors":["Mark Lewko"],"reviews":0,"subject":"math.NT","summary":"Let N(σ,T) count the nontrivial zeros of the Riemann zeta function with real part at least σ and imaginary part of absolute value at most T, counted with multiplicity. For every fixed σ ∈ [1/2,1] and every ε > 0, we prove N(σ,T) ≤ Cσ,ε T^(A(σ)(1−σ)+ε) for all T ≥ 2, where | σ interval | A(σ) | | --- | --- | | [1/2,9/13] | 3/(2−σ) | | (9/13,5/7] | 18/(7σ+3) | | (5/7,3/4] | 9/4 | | (3/4,1] | 3/(2σ) | The maximum of this coefficient bound is 39/17, improving the coefficient 30/13 in the Guth–Maynard bound. We also prove the separate estimate N(σ,T) ≤ Cε T^((39/17)(1−σ)+ε) for every ε > 0,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-11","reactions":{},"formalized":true,"formalizes":[{"title":"Selected Lean analytic proofs from Scott McColm repository, later pinned source","identifier":"https://github.com/smccolm/Lean/tree/6e2d10c6c2252ee1575bb7ef12dea12e2f1a3af1","relationship":"adapts"},{"title":"Selected Lean oscillatory, Abel/Jensen and contour proofs from earlier McColm pin","identifier":"https://github.com/smccolm/Lean/tree/2ace9e7c09a69fdcd1edae1ab6deb7cb3b4df1be","relationship":"adapts"},{"title":"Selected Fourier, Poisson, Mellin and Gamma proofs from prime-minor-arcs-2-15","identifier":"https://github.com/jconorgrogan/prime-minor-arcs-2-15/tree/f369f267b4dcfebf010c8e7baa2c9602e2960eba","relationship":"adapts"},{"title":"Selected PrimeNumberTheoremAnd analytic foundations, including attributed digamma source","identifier":"https://github.com/AlexKontorovich/PrimeNumberTheoremAnd/tree/4ecb950126c4290293c5662dfe0e884123171df5","relationship":"adapts"}]}