An explicit form of Ingham's zero density estimate

TL;DR

This paper presents an explicit version of Ingham's zero density estimate for the Riemann zeta-function, refining the logarithmic exponent, and provides an accurate estimate for the zeta-function's fourth power moment on the critical line.

Contribution

It offers an explicit form of Ingham's zero density estimate with a new logarithmic exponent and an asymptotically correct estimate for the zeta-function's fourth moment.

Findings

Explicit zero density estimate with improved logarithmic exponent

Asymptotically accurate estimate for the fourth power moment of zeta

Refinement of classical bounds on non-trivial zeros

Abstract

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