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On the Elementary Proof of the Prime Number Theorem

Published online by Cambridge University Press:  20 January 2009

N. Levinson
N. Levinson
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts
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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 2 , December 1966 , pp. 141 - 146
Copyright
Copyright © Edinburgh Mathematical Society 1966

References

REFERENCES

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