Explicit zero density for the Riemann zeta function

Let $N(σ,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $σ$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(σ,T)$ commonly referred to as a zero density result. In 1937, Ingham showed the following asymptotic result $N(σ,T)=\mathcal{O} ( T^{\frac83(1-σ)} (\log T)^5 )$. Ramaré 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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