Zero-density estimates from an improved large-values bound
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,…
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