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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

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.)


where is the characteristic function of prime numbers (A010051)
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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

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!

See also