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The Elementary Proof of the Prime Number Theorem: An Historical Perspective

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Number Theory

Abstract

The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x > 1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that

$$ \mathop {\lim }\limits_{x \to \infty } \pi \left( x \right){\raise0.7ex\hbox{${}$} \!\mathord{\left/ {\vphantom {{} {}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${}$}}\frac{x} {{\log (x)}} = 1. $$

This theorem was conjectured independently by Legendre and Gauss.

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Goldfeld, D. (2004). The Elementary Proof of the Prime Number Theorem: An Historical Perspective. In: Chudnovsky, D., Chudnovsky, G., Nathanson, M. (eds) Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9060-0_10

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  • DOI: https://doi.org/10.1007/978-1-4419-9060-0_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6490-3

  • Online ISBN: 978-1-4419-9060-0

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