In this paper, we prove a slight improvement of Huxley's zero-density estimate for $\zeta(s)$ near the line ${\rm Re}(s)=\frac{3}{4}$.
In this paper, two conjectures on the mean value of Dirichlet polynomials are given and are shown to imply good lower bound for $\int_H^{T+H}\vert\zeta(\frac{1}{2}+it)^k\vert^2\,dt$, uniform in $k$ and independent of $T$.
In this paper, we prove the Erdös conjecture that the binomial coefficient ${2n \choose n}$ is never square free, for all $n>4$.
