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:

\lambda(n) = (-1)^{\Omega(n)},\,\!

where Ω(n) is the number of prime factors of n, counted with multiplicity. (SIDN A008836).

λ is completely multiplicative since Ω(n) is additive. We have Ω(1) = 0 and therefore λ(1) = 1. The Liouville function satisfies the identity:

\sum_{d|n}\lambda(d)=1\,\! if n is a perfect square, and:
\sum_{d|n}\lambda(d)=0\,\! otherwise.

[edit] Series

The Dirichlet series for the Liouville function gives the Riemann zeta function as

\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.

The Lambert series for the Liouville function is

\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} = \sum_{n=1}^\infty q^{n^2} = \frac{1}{2}\left(\vartheta_3(q)-1\right),

where \vartheta_3(q) is the Jacobi theta function.

[edit] Conjectures

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 appearant scale invariance of the oscillations.

Logarithmic graph of the summatory Liouville function L(n) up to n=2\times10^9. The green bar shows the failure of the Pòlya conjecture; the blue curve shows the oscillatory contribution of the first Riemann zero.

Harmonic Summatory Liouville function M(n) up to n = 103

The Pólya conjecture is a conjecture made by George Pólya in 1919, stating that L(n) = \sum_{k=1}^n \lambda(k) \leq 0 for n>1. This turned out to be false. The smallest counter-example is n=906150257, found by Minoru Tanaka in 1980. It is not known as to whether L(n) changes sign infinitely often.

Defining the related sum M(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}, it was speculated for some time whether M(n) \geq 0 for sufficiently big n ≥ n0 (this "conjecture" is occasionally (but incorrectly) attributed to Paul Turan). This was then disproved by Haselgrove in 1958 (see the reference below), he showed that M(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 Paul Turan.

[edit] References

  1. Polya, G., Verschiedene Bemerkungen zur Zahlentheorie. Jahresbericht der deutschen Math.-Vereinigung 28 (1919), 31-40.
  2. Haselgrove, C.B. A disproof of a conjecture of Polya. Mathematika 5 (1958), 141-145.
  3. Lehman, R., On Liouville's function. Math. Comp. 14 (1960), 311-320.
  4. M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function. Tokyo Journal of Mathematics 3, 187-189, (1980).
  5. Eric W. Weisstein, Liouville Function at MathWorld.
  6. A.F. Lavrik (2001), “Liouville function”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104