Conjectures
The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining
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.
Read more about this topic: Liouville Function
Famous quotes containing the word conjectures:
“After all, it is putting a very high price on one’s conjectures to have a man roasted alive because of them.”
—Michel de Montaigne (1533–1592)
“Our conjectures pass upon us for truths; we will know what we do not know, and often, what we cannot know: so mortifying to our pride is the base suspicion of ignorance.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)