  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 $\kappa=1$.
