An explicit form of Ingham's zero density estimate

Ingham (1940) proved that $N(\sigma ,T)\ll T^{3(1-\sigma )\mathrm{/}(2-\sigma )}{\log }^{5}T$, where $N(\sigma ,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\mathrm{\Re }\{\rho \}\ge \sigma \ge 1\mathrm{/}2$ and $0<\mathrm{\Im }\{\rho \}\le T$. We provide an explicit version of this result with the exponent $(7-5\sigma )\mathrm{/}(2-\sigma )$ of the logarithmic factor. In addition, we also provide an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line.