Liouville function

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

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

$\lambda(n) = (-1)^{\Omega(n)},\,\!$

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

$\sum_{d|n}\lambda(d) = \begin{cases} 1 & \text{if }n\text{ is a perfect square,} \\ 0 & \text{otherwise.} \end{cases}$

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

$\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.$

The Lambert series for the Liouville function is

$\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} = \sum_{n=1}^\infty q^{n^2} = \frac{1}{2}\left(\vartheta_3(q)-1\right),$

where $\vartheta_3(q)$ is the Jacobi theta function.

Conjectures

File:Liouville.svg

Summatory Liouville function L(n) up to n = 10⁴. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.

File:Liouville-big.svg

Summatory Liouville function L(n) up to n = 10⁷. Note the apparent scale invariance of the oscillations.

File:Liouville-log.svg

Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10⁹. The green spike shows the function itself (not its negative) in the narrow region where the Pólya conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero.

File:Liouville-harmonic.svg

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

$L(n) = \sum_{k=1}^n \lambda(k),$

the conjecture states that $L(n)\leq 0$ 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 that L(n) < -1.3892783√n for infinitely many positive integers n.

Define the related sum

$T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.$

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

• Weisstein, Eric W., "Liouville Function", MathWorld.

• P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign Changes in Sums of the Liouville Function, Mathematics of Computation 77 (2008), no. 263, 1681–1694.