Updating the error term in the prime number theorem
Tóm tắt
An improved estimate is given for
$$|\theta (x) -x|$$
, where
$$\theta (x) = \sum _{p\le x} \log p$$
. 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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