Updating the error term in the prime number theorem

@article{Trudgian2014UpdatingTE,
  title={Updating the error term in the prime number theorem},
  author={Tim Trudgian},
  journal={The Ramanujan Journal},
  year={2014},
  volume={39},
  pages={225-234}
}
  • T. Trudgian
  • Published 2014
  • Mathematics
  • The Ramanujan Journal
An improved estimate is given for $$|\theta (x) -x|$$|θ(x)-x|, where $$\theta (x) = \sum _{p\le x} \log p$$θ(x)=∑p≤xlogp. Four applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, the third to a conjecture by Pomerance and the fourth to an inequality studied by Ramanujan. 

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