# Liouville function

From Encyclopedia of Mathematics

The arithmetic function defined by

where is the number of prime factors of . The Liouville function is closely connected with the Möbius function :

In number theory an important estimate is that of the sum

as . There is a conjecture that

The most recent result, obtained by a method of I.M. Vinogradov, has the form

The Liouville function was introduced by J. Liouville.

#### References

[1] | K. Prachar, "Primzahlverteilung" , Springer (1957) |

[2] | K. Chandrasekharan, "Arithmetical functions" , Springer (1970) |

**How to Cite This Entry:**

Liouville function. A.F. Lavrik (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Liouville_function&oldid=17862

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098