An explicit form of Ingham's zero density estimate

Abstract

Ingham (1940) proved that $N(\ddot{I}ƒ,T)\ll T^{3(1-\ddot{I}ƒ)/(2-\ddot{I}ƒ)}{\log }^{5}T$, where $N(\ddot{I}ƒ,T)$ counts the number of the non-trivial zeros $\ddot{I}$ of the Riemann zeta-function with $\Re \{\ddot{I}\}\ge \ddot{I}ƒ\ge 1/2$ and $0<\Im \{\ddot{I}\}\le T$. We provide an explicit version of this result with the exponent $(7-5\ddot{I}ƒ)/(2-\ddot{I}ƒ)$ 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.

31 pages, 1 Table; Comments are most welcome