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 additive. We have 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 Mobius function.

Series

The Dirichlet series for the Liouville function gives the Riemann zeta function as

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

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

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,^[1] 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 (but incorrectly) attributed to Pál Turán). This was then disproved by Haselgrove in 1958 (see the reference below), 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

Polya, G., Verschiedene Bemerkungen zur Zahlentheorie. Jahresbericht der deutschen Math.-Vereinigung 28 (1919), 31–40.
Haselgrove, C.B. A disproof of a conjecture of Polya. Mathematika 5 (1958), 141–145.
Lehman, R., On Liouville's function. Math. Comp. 14 (1960), 311–320.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function. Tokyo Journal of Mathematics 3, 187–189, (1980).
Weisstein, Eric W., "Liouville Function" from MathWorld.
A.F. Lavrik (2001), "Liouville function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=L/l059620

^ 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.