Introduction to Analytic Number Theory pp 74-105 | Cite as

# Some Elementary Theorems on the Distribution of Prime Numbers

Chapter

## 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
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## Copyright information

© Springer Science+Business Media New York 1976