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Liouville function

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Liouville function

The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.

Contents

If n is a positive integer, then λ(n) is defined as:

λ ( n ) = ( − 1 ) Ω ( n ) ,

where Ω(n) is the number of prime factors of n, counted with multiplicity (sequence A008836 in the OEIS).

λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). The number 1 has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity:

∑ d | n λ ( d ) = { 1 if  n  is a perfect square, 0 otherwise.

The Liouville function's Dirichlet inverse is the absolute value of the Möbius function.

Series

The Dirichlet series for the Liouville function is related to the Riemann zeta function by

ζ ( 2 s ) ζ ( s ) = ∑ n = 1 ∞ λ ( n ) n s .

The Lambert series for the Liouville function is

∑ n = 1 ∞ λ ( n ) q n 1 − q n = ∑ n = 1 ∞ q n 2 = 1 2 ( ϑ 3 ( q ) − 1 ) ,

where ϑ 3 ( q ) is the Jacobi theta function.

Conjectures

The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining

L ( n ) = ∑ k = 1 n λ ( k ) ,

the conjecture states that L ( n ) ≤ 0 for n > 1. This turned out to be false. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L(n) > 0.0618672√n for infinitely many positive integers n, while it can also be shown via the same methods that L(n) < -1.3892783√n for infinitely many positive integers n.

Define the related sum

T ( n ) = ∑ k = 1 n λ ( k ) k .

It was open for some time whether T(n) ≥ 0 for sufficiently big n ≥ n0 (this conjecture is occasionally–though incorrectly–attributed to Pál Turán). This was then disproved by Haselgrove (1958), who showed that T(n) takes negative values infinitely often. A confirmation of this positivity conjecture would have led to a proof of the Riemann hypothesis, as was shown by Pál Turán.

References

Liouville function Wikipedia
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