Möbius function
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]:
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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) |
Comments
The multiplicative arithmetic functions form a group under the convolution product . The Möbius function is in fact the inverse of the constant multiplicative function
(defined by
for all
) under this convolution product. From this there follows many "inversion formulas" , cf. e.g. Möbius series.
References
[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979) |
Möbius function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_function&oldid=30316