100% of the zeros of $\zeta(s)$ are on the critical line

Abstract

Throughout this manuscript the zeros are counted with multiplicity. We denote by $N(T)$ the number of zeros $\rho$ of $\zeta(s)$ in the critical strip upto height $T$ where $T>3$ is not an ordinate of zero of $\zeta(s)$. Denote by $N_0(T)$ the number of zeros $\rho$ of $\zeta(s)$ on the critical line upto height $T$. We first show that there exists $\epsilon_0>0$ such that $\xi(s)$ has no zeros on the boundary of a small rectangle $R_\epsilon$ defined as $R_\epsilon=\{\sigma+it\in\mathbb{C}\mid \frac{1}{2}-\epsilon\leq \sigma\leq \frac{1}{2}+\epsilon,\ 0\leq t\leq T\}$ whenever $0<\epsilon<\epsilon_0$. Secondly if $N_\epsilon(T)$ is the number of zeros $\rho$ of $\zeta(s)$ inside the rectangle $R_\epsilon$ then we prove that $N_\epsilon (T)=N_0(T)$ for $\epsilon$ sufficiently small depending on the height $T$. We use the Littlewood's lemma on the rectangle $R_\epsilon$ along with the Hadamard product of $\xi(s)$ and the asymptotic for the logarithmic derivative of $\zeta(s)$ to prove that as $T\to \infty$, $$N_0(T)=\frac{T}{2\pi}\log\left(\frac{T}{2\pi}\right)-\frac{T}{2\pi}+\mathcal{O}(\log T)$$ Also if $\kappa$ is the proportion of zeros of $\zeta(s)$ on the critical line $$\kappa:=\liminf_{T\to \infty} \frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $\kappa=1$.