# Liouville function

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.

Liouville function satisfies the explicit formula $$ \sum_{n=1}^\infty \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\rho}\frac{h( \gamma)\zeta(2 \rho )}{\zeta'( \rho)} + \frac{1}{\zeta (1/2)}\int_{-\infty}^\infty dx \, g(x) $$

where $ g(u)= \frac{1}{2\pi} \int_{-\infty}^\infty h(x)\exp(-iux) $ form a Fourier transform pair

#### References

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

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

**How to Cite This Entry:**

Liouville function.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Liouville_function&oldid=44333