If n is a positive integer, then λ(n) is defined as:
λ is completely multiplicative since Ω(n) is completely additive, i.e.: Ω(ab) = Ω(a) + Ω(b). The number one has no prime factors, so Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity:
The Lambert series for the Liouville function is
where is the Jacobi theta function.
the conjecture states that 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
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.
- 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. B. (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, M. (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function". Tokyo Journal of Mathematics 3: 187–189.
- 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