Explicit zero density for the Riemann zeta function

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 $N(\sigma ,T)=\mathcal{O}(T^{\frac{8}{3}(1-\sigma )}(\log T{)}^5)$. 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.

24 pages