Liouville function

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. ( [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008836 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.

eries

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

:$sum\_\{n=1\}^infty\; frac\{lambda(n)q^n\}\{1-q^n\}\; =\; sum\_\{n=1\}^infty\; q^\{n^2\}\; =\; frac\{1\}\{2\}left(vartheta\_3(q)-1\; ight),$

where $vartheta\_3(q)$ is the Jacobi theta function.

Conjectures

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 Pál Turán). 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 Pál Turán.

References

# Polya, G., "Verschiedene Bemerkungen zur Zahlentheorie." Jahresbericht der deutschen Math.-Vereinigung 28 (1919), 31-40.
# Haselgrove, C.B. "A disproof of a conjecture of Polya." Mathematika 5 (1958), 141-145.
# Lehman, R., "On Liouville's function." Math. Comp. 14 (1960), 311-320.
# M. Tanaka, "A Numerical Investigation on Cumulative Sum of the Liouville Function." Tokyo Journal of Mathematics 3, 187-189, (1980).
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