Prime-counting function

This article uses technical mathematical notation for logarithms. All instances of

log(x)

without a subscript base should be interpreted as a natural logarithm, also commonly written as

ln(x)

or

log e (x)

.

The values of $π (n)$ for the first 60 positive integers

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$ .^[1]^[2] It is denoted by $π (x)$ (unrelated to the number $π$ ).

A symmetric variant seen sometimes is $π 0 (x)$ , which is equal to $π(x) − .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄2$ if $x$ is exactly a prime number, and equal to $π (x)$ otherwise. That is, the number of prime numbers less than $x$ , plus half if $x$ equals a prime.

^ Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8. ISBN 0-262-02405-5.
^ Weisstein, Eric W. "Prime Counting Function". MathWorld.

[Bach-1] Bach, Eric; Shallit, Jeffrey (1996). Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8. ISBN 0-262-02405-5.

[mathworld_pcf-2] Weisstein, Eric W. "Prime Counting Function". MathWorld.

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