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The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.
Contents
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).
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 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
The Lambert series for the Liouville function is
where
Conjectures
The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining
the conjecture states that
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.