For a broad class of zero-free regions, the paper sharpens the prime number theorem error term and constructs Beurling systems showing this sharpening is nearly optimal.
Explicit zero density estimate near unity
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abstract
We will provide the first explicit zero-density estimate for $\zeta$ of the form $N(\sigma,T)\le \mathcal{C}T^{B(1-\sigma)^{3/2}}(\log T)^C$. In particular, we improve $C$ to $10393/900=11.547\dots.$
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On the connection between zero-free regions and the error term in the Prime Number Theorem
For a broad class of zero-free regions, the paper sharpens the prime number theorem error term and constructs Beurling systems showing this sharpening is nearly optimal.