Explicit zero density for the Riemann zeta function

Habiba Kadiri; Allysa Lumley; Nathan Ng

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arXiv · 2101.12263

Explicit zero density for the Riemann zeta function

Habiba Kadiri·Allysa Lumley·Nathan Ng

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Abstract

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(\sigma,T)=\mathcal{O} ( T^{\frac83(1-\sigma)} (\log T)^5 )$. Ramar\'{e} recently proved an explicit version of this estimate. We discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants.

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Habiba Kadiri, Allysa Lumley, Nathan Ng. 2021-01-28. Explicit zero density for the Riemann zeta function. https://arxiv.org/abs/2101.12263

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