  In this paper we first establish new explicit estimates for Chebyshev's
$\vartheta$-function. Applying these new estimates, we derive new upper and
lower bounds for some functions defined over the prime numbers, for instance
the prime counting function $\pi(x)$, which improve the currently best ones.
Furthermore, we use the obtained estimates for the prime counting function to
give two new results concerning the existence of prime numbers in short
intervals.
