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Some Elementary Theorems on the Distribution of Prime Numbers

  • Tom M. Apostol
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

If x > 0 let π(x) denote the number of primes not exceeding x. Then π(x) → ∞ as x → ∞ since there are infinitely many primes. The behavior of π(x)as a function of x has been the object of intense study by many celebrated mathematicians ever since the ighteenth century. Inspection of tables of primes led Gauss (1792) and Legendre (1798) to conjecture that π(x) is asymptotic to x/log x, that is
$$\mathop {\lim }\limits_{x \to \infty } \frac{{\pi \left( x \right)\log x}}{x} = 1. $$

Keywords

Prime Number Tauberian Theorem Arithmetical Function Prime Number Theorem Elementary Theorem 
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 Science+Business Media New York 1976

Authors and Affiliations

  • Tom M. Apostol
    • 1
  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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