If n is a positive integer, then λ(n) is defined as:
where Ω(n) is the number of prime factors of n, counted with multiplicity (sequence A008836 in the OEIS). If n is squarefree, i.e., if where is prime for all i and where , then we have the following alternate formula for the function expressed in terms of the Möbius function and the distinct prime factor counting function :
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 Dirichlet inverse of Liouville function is the absolute value of the Möbius function, which is equivalently the characteristic function of the squarefree integers. We also have that , and that for all natural numbers n:
The Lambert series for the Liouville function is
where is the Jacobi theta function.
Conjectures on weighted summatory functions
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.
For any , assuming the Riemann hypothesis, we have that the summatory function is bounded by
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.
These -weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Moebius function. In fact, we have that the so-termed non-weigthed, or ordinary function precisely corresponds to the sum
which then can be inverted via the inverse transform to show that for , and
where we can take , and with the remainder terms defined such that and as .
In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by , of the Riemann zeta function are simple, then for any and there exists an infinite sequence of which satisfies that for all v such that
where for any increasingly small we define
and where the remainder term
which of course tends to 0 as . These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since we have another similarity in the form of to in so much as the dominant leading term in the previous formulas predicts a negative bias in the values of these functions over the positive natural numbers x.
- 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.
- Humphries, Peter (2013). "The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture". Journal of Number Theory. 133 (2): 545–582. arXiv:1108.1524. doi:10.1016/j.jnt.2012.08.011.
- 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 (72): 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 (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4