Liouville function

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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.


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098