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The prime number theorem and Selberg’s method

  • K. Chandrasekharan
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 167)

Abstract

Let π(x) denote, for any real x, the number of primes not exceeding x. The prime number theorem is the assertion that
$$ \mathop{{\lim}}\limits_{{x\to\infty}}(\frac{{\pi (x)}}{{x/\log x}})=1 $$
(1)
A fundamental formula discovered by Atle Selberg has made a proof of (1) possible without the use of the properties of the zeta-function of Riemann, and without the use of the theory of functions of a complex variable. We shall prove Selberg’s formula in this chapter, and indicate some of its consequences. We shall also prove an inequality due to E. Wirsing, which, when combined with a variant of Selberg’s formula, gives a proof of the prime number theorem.

Keywords

Arithmetical Progression Arithmetical Function Prime Number Theorem Elementary Argument Divisor Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1970

Authors and Affiliations

  • K. Chandrasekharan
    • 1
  1. 1.Eidgenössische Technische Hochschule ZürichDeutschland

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