In number theory, the Liouville function, named after French mathematician Joseph Liouville and denoted <math>\lambda(n)</math>, is an important arithmetic function. Its value is <math>1</math> if <math>n</math> is the product of an even number of prime numbers, and <math>-1</math> if it is the product of an odd number of prime numbers.
Definition
By the fundamental theorem of arithmetic, any positive integer <math>n</math> can be represented uniquely as a product of powers of primes:
:<math>n=p_1^{a_1}\cdots p_k^{a_k}</math>,
where <math>p_1,\dots,p_k</math> are primes and the exponents <math>a_1,\dots,a_k</math> are positive integers. The prime omega function <math>\Omega(n)</math> counts the number of primes in the factorization of <math>n</math> with multiplicity:
:<math>\Omega(n) = a_1 + a_2 + \cdots + a_k</math>.
Thus, the Liouville function is defined by
: <math> \lambda(n) = (-1)^{\Omega(n)}</math>
.
Properties
Since <math>\Omega(n)</math> is completely additive; i.e., <math>\Omega(ab)=\Omega(a)+\Omega(b)</math>, then <math>\lambda(n)</math> is completely multiplicative. Since <math>1</math> has no prime factors, <math>\Omega(1)=0</math>, so <math>\lambda(1)=1</math>.
<math>\lambda(n)</math> is also related to the Möbius function <math>\mu(n)</math>: if we write <math>n</math> as <math>n=a^2b</math>, where <math>b</math> is squarefree, then
: <math> \lambda(n) = \mu(b). </math>
The sum of the Liouville function over the divisors of <math>n</math> is the characteristic function of the squares:
:<math>
\sum_{d|n}\lambda(d) =
\begin{cases}
1 & \text{if }n\text{ is a perfect square,} \\
0 & \text{otherwise.}
\end{cases}
</math>
Möbius inversion of this formula yields
:<math>\lambda(n) = \sum_{d^2|n} \mu\left(\frac{n}{d^2}\right).</math>
The Dirichlet inverse of the Liouville function is the absolute value of the Möbius function, <math>\lambda^{-1}(n)=|\mu(n)|=\mu^2(n)</math>, the characteristic function of the squarefree integers.
Series
The Dirichlet series for the Liouville function is related to the Riemann zeta function by
:<math>\frac{\zeta(2s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\lambda(n)}{n^s}.</math>
Also:
:<math>\sum\limits_{n=1}^{\infty} \frac{\lambda(n)\ln n}{n}=-\zeta(2)=-\frac{\pi^2}{6}.</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} =
\frac{1}{2}\left(\vartheta_3(q)-1\right),</math>
where <math>\vartheta_3(q)</math> is the Jacobi theta function.
Conjectures on weighted summatory functions
<div style="float: right; clear: right">
thumb|none|Summatory Liouville function L(n) up to n = 10<sup>4</sup>. The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.
thumb|none|Summatory Liouville function L(n) up to n = 10<sup>7</sup>. Note the apparent [[scale invariance of the oscillations.]]
thumb|none|Logarithmic graph of the negative of the summatory Liouville function L(n) up to n = 2 × 10<sup>9</sup>. 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.]]
thumb|none|Harmonic Summatory Liouville function T(n) up to n = 10<sup>3</sup>
</div>
The Pólya problem is a question raised made by George Pólya in 1919. Defining
: <math>L(n) = \sum_{k=1}^n \lambda(k)</math> ,
the problem asks whether L(n) ≤ 0 for all n > 1. The answer turns out to be no. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L(n) > 0.0618672 for infinitely many positive integers n, while it can also be shown via the same methods that L(n) < −1.3892783 for infinitely many positive integers n.
Define the related sum
: <math>T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.</math>
It was open for some time whether T(n) ≥ 0 for sufficiently big n ≥ n<sub>0</sub> (this conjecture is occasionally—though incorrectly—attributed to Pál Turán). This was then disproved by , 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.
Generalizations
More generally, we can consider the weighted summatory functions over the Liouville function defined for any <math>\alpha \in \mathbb{R}</math> as follows for positive integers x where (as above) we have the special cases <math>L(x) := L_0(x)</math> and <math>T(x) = L_1(x)</math>