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Liouville Function

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LiouvilleLambda

The Liouville function is defined by

lambda(n)=(-1)^(Omega(n)),
(1)

where n is a positive integer and Omega(n) is the number of not necessarily distinct prime factors of n, with Omega(1)=0. The values of lambda(n) for n=1, 2, ... are 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, ... (OEIS A008836). The values of n such that lambda(n)=-1 are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, ... (OEIS A026424), while the values such that lambda(n)=+1 are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).

The Liouville function is implemented in the Wolfram Language as LiouvilleLambda[n].

The Liouville function is connected with the Riemann zeta function by the equation

(zeta(2s))/(zeta(s))=sum_(n=1)^infty(lambda(n))/(n^s)
(2)

for Re[s]>1 (Lehman 1960). It has the Lambert series

sum_(n=1)^(infty)(lambda(n)x^n)/(1-x^n)=sum_(n=1)^(infty)x^(n^2)
(3)
=1/2[theta_3(x)-1],
(4)

where |x|<1 and theta_3(x)=theta_3(0,x) is a Jacobi theta function.

The associated Liouville summatory function is denoted L(n).

See also

Liouville Summatory Function, Pólya Conjecture, Prime Factor, Prime Omega Function, Riemann Zeta Function

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References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, p. 37, 1976.Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.Sloane, N. J. A. Sequences A008836, A026424, and A028260 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Liouville Function

Cite this as:

Weisstein, Eric W. "Liouville Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LiouvilleFunction.html

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