Explicit estimates of some functions over primes

  title={Explicit estimates of some functions over primes},
  author={Pierre Dusart},
  journal={The Ramanujan Journal},
  • P. Dusart
  • Published 2018
  • Mathematics
  • The Ramanujan Journal
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 $$\zeta $$ζ zeros like $$\psi (x),\vartheta (x),\pi (x)$$ψ(x),ϑ(x),π(x), or the $$k\mathrm{{th}}$$kth prime number $$p_k$$pk. 
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