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

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.

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

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.

Harmonic Summatory Liouville function T(n) up to n = 10³

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–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

1. 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.

• Polya, G. (1919). "Verschiedene Bemerkungen zur Zahlentheorie". Jahresbericht der Deutschen Mathematiker-Vereinigung 28: 31–40.
• Haselgrove, C. Brian (1958). "A disproof of a conjecture of Polya". Mathematika 5 (2): 141–145. doi:10.1112/S0025579300001480. ISSN 0025-5793. MR 0104638. Zbl 0085.27102.
• Lehman, R. (1960). "On Liouville's function". Math. Comp. 14: 311–320. doi:10.1090/S0025-5718-1960-0120198-5. MR 0120198.
• Tanaka, Minoru (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function". Tokyo Journal of Mathematics 3 (1): 187–189. doi:10.3836/tjm/1270216093. MR 0584557.
• Weisstein, Eric W., "Liouville Function", MathWorld.
• A.F. Lavrik (2001), "Liouville function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4