Abstract
Perhaps the simplest elementary proof of the prime number theorem, see Erdös (2) and Selberg (5), is Wright's modification (8), (3, p. 362) of Selberg's original proof (5). Another variant is due to V. Nevanlinna (4). Wright's proof uses Selberg's idea of smoothing the weighting process which occurs in the Selberg inequality, (1.2) below, by iterating this inequality. Here it will be shown that the proof requires less ingenuity if use is made of a further smoothing operation, namely first integrating the Selberg inequality itself. Integration has been used on a related inequality by Breusch (1 to obtain a remainder term. This method also makes proof by contradiction unnecessary.
Publisher
Cambridge University Press (CUP)
Reference8 articles.
1. The elementary proof of the prime number theorem;Wright;Proc. Roy. Soc. Edinburgh,1951
2. Über den elementaren Beweis des Primzahlsatzes;Mevanlinna;Soc. Sci. Fenn. Comment. Phys.-Math,1962
3. An elementary proof of the prime number theorem with remainder term
4. On a New Method in Elementary Number Theory Which Leads to An Elementary Proof of the Prime Number Theorem
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