Updating the error term in the prime number theorem

Tim Trudgian

doi:10.1007/s11139-014-9656-6

Updating the error term in the prime number theorem

Tim Trudgian^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

34 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	225-234
Number of pages	10
Journal	Ramanujan Journal
Volume	39
Issue number	2
DOIs	https://doi.org/10.1007/s11139-014-9656-6
Publication status	Published - 1 Feb 2016

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title = "Updating the error term in the prime number theorem",

abstract = "An improved estimate is given for \$\$|\textbackslash{}theta (x) -x|\$\$|θ(x)-x|, where \$\$\textbackslash{}theta (x) = \textbackslash{}sum \_\{p\textbackslash{}le x\} \textbackslash{}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.",

keywords = "Chebyshev functions, Prime number theorem, Ramanujan{\textquoteright}s inequality",

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AB - 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.

KW - Chebyshev functions

KW - Prime number theorem

KW - Ramanujan’s inequality

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JO - Ramanujan Journal

JF - Ramanujan Journal

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ER -

Updating the error term in the prime number theorem

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