An explicit form of Ingham's zero density estimate
Shashi Chourasiya, Aleksander Simonič
TL;DR
This work delivers fully explicit zero-density bounds for the non-trivial zeros of the Riemann zeta-function and explicit, main-term accurate bounds for the fourth power moment on the critical line. The authors fuse Ingham-type zero-density arguments with Gabriel's convexity framework and precise Dirichlet-polynomial mean-value estimates (via mollifiers $f_X$) to obtain explicit constants and intervales for $\sigma$ and $T$. A Ramachandra-style analysis yields an explicit fourth moment formula with a clearly identified main term $\frac{1}{\pi^2}T\log^{4}(T/2)$ and quantitatively controlled error, including an explicit bound for related moments like $\mathcal{M}_{3/2}(T)$. An appendix provides an explicit improvement of Kadiri's zero-density bound, enabling sharper explicit ranges and constants. The results have implications for explicit bounds in the prime-number distribution and divisor problems, and the techniques may extend to other $L$-functions via the mollification and convexity framework.
Abstract
Ingham (1940) proved that $N(σ,T)\ll T^{3(1-σ)/(2-σ)}\log^{5}{T}$, where $N(σ,T)$ counts the number of the non-trivial zeros $ρ$ of the Riemann zeta-function with $\Re\{ρ\}\geqσ\geq 1/2$ and $0<\Im\{ρ\}\leq T$. We provide an explicit version of this result with the exponent $(7-5σ)/(2-σ)$ 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.