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Characteristic function of prime numbers
The characteristic function of prime numbers, which gives 1 if the number is a prime number and 0 otherwise, is defined as
where is the prime counting function.
It is also given by
where is the greatest common divisor of and , is the primorial of and is the Iverson bracket.
For example, , . (See A010051 for a listing going up to 100000.)
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Asymptotic probability
The characteristic function of prime numbers is equal to 1 with the slowly decreasing (converging towards 0) asymptotic probability
and equal to 0 with the slowly increasing (converging towards 1) asymptotic probability
Observe that if you set to = 2.718281828459..., giving a probability of being prime equal to 1, we have that both integers surrounding , 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
where is the th prime, happens to diverge towards infinity!