Sharper estimates for Chebyshev's functions $\vartheta$ and $ψ$

In this article we present some improved results for Chebyshev's functions $\vartheta$ and $ψ$ 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.

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