  In this paper we give effective estimates for some classical arithmetic
functions defined over prime numbers. First we find the smallest real number
$x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds
for every $x \geq x_0$. Then we give some new results concerning the existence
of prime numbers in short intervals. Also we derive new upper and lower bounds
for some functions defined over prime numbers, for instance the prime counting
function $\pi(x)$, which improve current best estimates of similar shape.
