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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.

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
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article