Abstract
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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Notes
There is also the result of Ford [7]
$$\begin{aligned} \pi (x) - \text {li}(x) = O( x \exp \{-0.2098 (\log x)^{3/5} (\log \log x)^{-1/5}\}). \end{aligned}$$It appears that this result has not been made explicit.
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Acknowledgments
I am grateful to Szymon Brzostowski for verifying one of the computations leading to Corollary 2.
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Dedicated to MG Johnson, RJ Harris, PM Siddle and NM Lyon, all of whom enabled me to work two extra days on this article.
This article was supported by Australian Research Council DECRA Grant DE120100173.
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Trudgian, T. Updating the error term in the prime number theorem. Ramanujan J 39, 225–234 (2016). https://doi.org/10.1007/s11139-014-9656-6
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DOI: https://doi.org/10.1007/s11139-014-9656-6