Characteristic function of prime numbers

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The characteristic function of prime numbers, which gives 1 if the number is a prime number and 0 otherwise, is defined as

$\chi _{\rm {prime}}(n)\equiv \pi (n)-\pi (n-1)\,$

where $\scriptstyle \pi (n)\,$ is the prime counting function.

It is also given by

$\chi _{\rm {prime}}(n)=[\gcd(n,\lfloor {\sqrt {n}}\rfloor \#)=1],\,$

where $\scriptstyle \gcd(m,\,n)\,$ is the greatest common divisor of $\scriptstyle m\,$ and $\scriptstyle n\,$, $\scriptstyle n\#\,$ is the primorial of $\scriptstyle n\,$ and $\scriptstyle [\cdot ]\,$ is the Iverson bracket.

For example, $\scriptstyle \chi _{\rm {prime}}(7)\,=\,1\,$, $\scriptstyle \chi _{\rm {prime}}(8)\,=\,0\,$. (See A010051 for a listing going up to 100000.)

$\scriptstyle \chi _{\rm {prime}}(n),\,1\,\leq \,n\,\leq \,257,\,$ where $\scriptstyle \chi _{\rm {prime}}(n)\,$ is the characteristic function of prime numbers (A010051)

1     $\scriptstyle \chi _{\rm {prime}}(n)\,$
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0

Asymptotic probability

The characteristic function of prime numbers is equal to 1 with the slowly decreasing (converging towards 0) asymptotic probability

$\lim _{n\to \infty }P(\chi _{\rm {prime}}(n)=1)={\frac {1}{\log(n)}}={\frac {\log(e)}{\log(n)}},\,$

and equal to 0 with the slowly increasing (converging towards 1) asymptotic probability

$\lim _{n\to \infty }P(\chi _{\rm {prime}}(n)=0)=1-{\frac {1}{\log(n)}}={\frac {\log(n)-1}{\log(n)}}={\frac {\log(n/e)}{\log(n)}}.\,$

Observe that if you set $\scriptstyle n\,$ to $\scriptstyle e\,$ = 2.718281828459..., giving a probability of $\scriptstyle n\,$ being prime equal to 1, we have that both integers surrounding $\scriptstyle e\,$, i.e. 2 and 3, are prime!

The above limits imply that 0% of the integers are prime (100% of the integers are composite), yet the infinite series

$\sum _{n=1}^{\infty }{\frac {1}{p_{n}}}=\sum _{n=1}^{\infty }[\chi _{\rm {prime}}(n)=1]\,{\frac {1}{n}}=\sum _{n=1}^{\infty }[\gcd(n,\lfloor {\sqrt {n}}\rfloor \#)=1]\,{\frac {1}{n}}\to \infty ,\,$

where $\scriptstyle p_{n}\,$ is the $\scriptstyle n\,$th prime, happens to diverge towards infinity!

See also

• Prime counting function
• Characteristic functions