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Distribution of Primes

The study of the distribution of prime numbers among the integers.

Sequence of Expressions

Theorem

Proves the PNT using only basic complex analysis and Tauberian theorems.

Theorem

The number of primes

\le x

in the progression

a, a+q, a+2q, \dots

is asymptotic to

\frac{1}{\phi(q)} \frac{x}{\ln x}

Theorem

Advanced

The number of primes less than or equal to

x

is approximately

x / \ln(x)

.\n

\pi(x) \sim \frac{x}{\ln(x)}