# Möbius function

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An arithmetic function of natural argument: , if is divisible by the square of a prime number, otherwise , where is the number of prime factors of the number . This function was introduced by A. Möbius in 1832.

The Möbius function is a multiplicative arithmetic function; if . It is used in the study of other arithmetic functions; it appears in inversion formulas (see, e.g. Möbius series). The following estimate is known for the mean value of the Möbius function [2]:

where is a constant. The fact that the mean value tends to zero as implies an asymptotic law for the distribution of prime numbers in the natural series.

#### References

 [1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) [2] A. Walfisz, "Weylsche Exponentialsummen in der neueren Zahlentheorie" , Deutsch. Verlag Wissenschaft. (1963)