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)=O(T83(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.