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:

<math>\lambda(n) = (-1)^{\Omega(n)}\,\! </math>,

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:

<math>\sum_{d|n}\lambda(d)=1\,\! </math> if n is a perfect square, and:

<math>\sum_{d|n}\lambda(d)=0\,\! </math> otherwise.

The Liouville function is related to the Riemann zeta function by the formula

<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}</math>

The Lambert series for the Liouville function is

<math>\sum_{n=1}^\infty \frac{\lambda(n)q^n}{1-q^n} =

\sum_{n=1}^\infty q^{n^2}</math>

with the sum on the left similar to the Ramanujan theta function.

Conjectures

Polya conjectured that <math>L(n) = \sum_{k=1}^n \lambda(k) \leq 0 </math> for n>1. This turned out to be false, n=906150257 being a counterexample (found by Minoru Tanaka in 1980). It is not known as to whether L(n) changes sign infinitely often.

Also, if we define <math>M(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}</math>, it was speculated for some time whether <math>M(n) \geq 0 </math> 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.

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).fr:Fonction de Liouville

it:Funzione di Liouville sv:Liouvilles lambda-funktion