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 $R_{\epsilon }=\{\sigma +it\in \mathbb{C}\mid \frac{1}{2}-\epsilon \le \sigma \le \frac{1}{2}+\epsilon ,0\le t\le T\}$ 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 $$\kappa :=\underset{T\to \infty }{\mathrm{liminf}}\frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $\kappa =1$.

Not worthy of publication. Needs modification