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The prime number theorem:Analytic and elementary proofs

Ciarán O’Rourke-2013-02-01-MURAL - Maynooth University Research Archive Library (National University of Ireland, Maynooth)
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TL;DRAbstract

Three proofs of the prime number theorem are presented. The �rst is a heavily analytic proof based on early accounts. Cauchy's residue theorem and various results relating to the Riemann zeta function play a vital role. A weaker result than the prime number theorem is used for the proof, namely Chebyshev's theorem. The second proof is elementary in the sense that it involves no complex analysis. Instead, mainly number-theoretic results are used, in particular, Selberg's formulas. The third proof, like the �rst, relies heavily on the Riemann zeta function, but is considerably shorter for the use of the Laplace transform and the analytic theorem.

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Three proofs of the prime number theorem are presented. The �rst is a heavily analytic proof based on early accounts. Cauchy's residue theorem and various results relating to the Riemann zeta function play a vital role. A weaker result than the prime number theorem is used for the proof, namely Chebyshev's theorem. The second proof is elementary in the sense that it involves no complex analysis. Instead, mainly number-theoretic results are used, in particular, Selberg's formulas. The third proof, like the �rst, relies heavily on the Riemann zeta function, but is considerably shorter for the use of the Laplace transform and the analytic theorem.

Keywords

MathematicsPrime number theoremAnalytic number theoryMathematical proofAnalytic proofElementary proofMultiplicative number theoryRiemann zeta function

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