Explicit estimates of some functions over primes.

New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann hypothesis implies results about the distribution of prime numbers. We get better effective estimates of common number theoretical functions which are closely linked to $$\zeta $$ zeros like $$\psi (x),\vartheta (x),\pi (x)$$ , or the $$k\mathrm{{th}}$$ prime number $$p_k$$ . [ABSTRACT FROM AUTHOR]