Some Elementary Theorems on the Distribution of Prime Numbers

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


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 eighteenth 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,
$$\mathop {\lim }\limits_{x \to \infty } \frac{{\pi (x)\log x}}{x} = 1.$$


Prime Number Asymptotic Formula Tauberian Theorem Prime Number Theorem Elementary Theorem 
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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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