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 1 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√ for infinitely many positive integers n, while it can also be shown via the same methods that L(n) < -1.3892783√ 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–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.
- 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.
- Peter Humphries, The distribution of weighted sums of the Liouville function and Pólya’s conjecture, Journal of Number Theory 133 (2013), 545–582.
- 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 Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4