  In this article, we prove an explicit bound for $N(\sigma,T)$, the number of
zeros of the Riemann zeta function satisfying $\sigma < \Re s <1 $ and $0 < \Im
s < T$. This result provides a significant improvement over Rosser's bound for
$N(T)$ when used for estimating prime counting functions. For instance this is
applied to obtain new bounds for $\psi(x)$ (arXiv:1310.6374).
