  In this article we present some improved results for Chebyshev's functions
$\vartheta$ and $\psi$ using the new zero-free region obtained by H. Kadiri and
the calculated the first $10^{13}$ zeros of the Riemann zeta function on the
critical line by Xavier Gourdon. The methods in the proofs are similar to those
of Rosser-Shoenfeld papers on this subject.
