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Difference between revisions of "Liouville function"

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The Liouville function was introduced by J. Liouville.
 
The Liouville function was introduced by J. Liouville.
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Liouville function satisfies the explicit formula $$ \sum_{n=1}^\infty \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\rho}\frac{h( \gamma)\zeta(2 \rho )}{\zeta'( \rho)} + \frac{1}{\zeta (1/2)}\int_{-\infty}^\infty dx \, g(x) $$
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where  $ g(u)= \frac{1}{2\pi} \int_{-\infty}^\infty h(x)\exp(-iux) $ form a Fourier transform pair
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Chandrasekharan,  "Arithmetical functions" , Springer  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Chandrasekharan,  "Arithmetical functions" , Springer  (1970)</TD></TR></table>

Latest revision as of 11:53, 17 January 2020

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://encyclopediaofmath.org/index.php?title=Liouville_function&oldid=17862
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article