Liouville function

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The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.

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[edit]

The Dirichlet series for the Liouville function is related to the Riemann zeta function by

The Lambert series for the Liouville function is

where is the Jacobi theta function.

Conjectures[edit]

Summatory Liouville function L(n) up to n = 104. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.
Summatory Liouville function L(n) up to n = 107. Note the apparent scale invariance of the oscillations.
Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 109. The green spike shows the function itself (not its negative) in the narrow region where the Pólya conjecture fails; the blue curve shows the oscillatory contribution of the first Riemann zero.
Harmonic Summatory Liouville function T(n) up to n = 103

The Pólya conjecture is a conjecture made by George Pólya in 1919. Defining

(sequence A002819 in the OEIS),

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) > 0.0618672n for infinitely many positive integers n,[1] while it can also be shown via the same methods that L(n) < -1.3892783n for infinitely many positive integers n.[2]

Define the related sum

It was open for some time whether T(n) ≥ 0 for sufficiently big nn0 (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.

References[edit]

  1. ^ 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.
  2. ^ Peter Humphries, The distribution of weighted sums of the Liouville function and Pólya’s conjecture, Journal of Number Theory 133 (2013), 545–582.