Liouville function
Template:Short description 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
The Pólya problem is a question raised made by George Pólya in 1919. Defining
- <math>L(n) = \sum_{k=1}^n \lambda(k)</math> Template:OEIS,
the problem asks whether <math>L(n)\leq 0</math> for some n > 1. The answer turns out to be yes. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L(n) > 0.0618672Template:Radic for infinitely many positive integers n,<ref>Template:Cite journal</ref> while it can also be shown via the same methods that L(n) < −1.3892783Template:Radic for infinitely many positive integers n.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS" />
For any <math>\varepsilon > 0</math>, assuming the Riemann hypothesis, we have that the summatory function <math>L(x) \equiv L_0(x)</math> is bounded by
- <math>L(x) = O\left(\sqrt{x} \exp\left(C \cdot \log^{1/2}(x) \left(\log\log x\right)^{5/2+\varepsilon}\right)\right),</math>
where the <math>C > 0</math> is some absolute limiting constant.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS">Template:Cite journal</ref>
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 ≥ n0 (this conjecture is occasionally—though incorrectly—attributed to Pál Turán). This was then disproved by Template:Harvtxt, 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> <ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/>
- <math>L_{\alpha}(x) := \sum_{n \leq x} \frac{\lambda(n)}{n^{\alpha}}.</math>
These <math>\alpha^{-1}</math>-weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Möbius function. In fact, we have that the so-termed non-weighted, or ordinary, function <math>L(x)</math> precisely corresponds to the sum
- <math>L(x) = \sum_{d^2 \leq x} M\left(\frac{x}{d^2}\right) = \sum_{d^2 \leq x} \sum_{n \leq \frac{x}{d^2}} \mu(n).</math>
Moreover, these functions satisfy similar bounding asymptotic relations.<ref name="HUMPHRIES-WEIGHTED-SUMFUNCS"/> For example, whenever <math>0 \leq \alpha \leq \frac{1}{2}</math>, we see that there exists an absolute constant <math>C_{\alpha} > 0</math> such that
- <math>L_{\alpha}(x) = O\left(x^{1-\alpha}\exp\left(-C_{\alpha} \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).</math>
By an application of Perron's formula, or equivalently by a key (inverse) Mellin transform, we have that
- <math>\frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} = s \cdot \int_1^{\infty} \frac{L_{\alpha}(x)}{x^{s+1}} dx,</math>
which then can be inverted via the inverse transform to show that for <math>x > 1</math>, <math>T \geq 1</math> and <math>0 \leq \alpha < \frac{1}{2}</math>
- <math>L_{\alpha}(x) = \frac{1}{2\pi\imath} \int_{\sigma_0-\imath T}^{\sigma_0+\imath T} \frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)}
\cdot \frac{x^s}{s} ds + E_{\alpha}(x) + R_{\alpha}(x, T), </math>
where we can take <math>\sigma_0 := 1-\alpha+1 / \log(x)</math>, and with the remainder terms defined such that <math>E_{\alpha}(x) = O(x^{-\alpha})</math> and <math>R_{\alpha}(x, T) \rightarrow 0</math> as <math>T \rightarrow \infty</math>.
In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by <math>\rho = \frac{1}{2} + \imath\gamma</math>, of the Riemann zeta function are simple, then for any <math>0 \leq \alpha < \frac{1}{2}</math> and <math> x \geq 1</math> there exists an infinite sequence of <math>\{T_v\}_{v \geq 1}</math> which satisfies that <math>v \leq T_v \leq v+1</math> for all v such that
- <math>L_{\alpha}(x) = \frac{x^{1/2-\alpha}}{(1-2\alpha) \zeta(1/2)} + \sum_{|\gamma| < T_v} \frac{\zeta(2\rho)}{\zeta^{\prime}(\rho)} \cdot
\frac{x^{\rho-\alpha}}{(\rho-\alpha)} + E_{\alpha}(x) + R_{\alpha}(x, T_v) + I_{\alpha}(x), </math>
where for any increasingly small <math>0 < \varepsilon < \frac{1}{2}-\alpha</math> we define
- <math>I_{\alpha}(x) := \frac{1}{2\pi\imath \cdot x^{\alpha}} \int_{\varepsilon+\alpha-\imath\infty}^{\varepsilon+\alpha+\imath\infty}
\frac{\zeta(2s)}{\zeta(s)} \cdot \frac{x^s}{(s-\alpha)} ds,</math>
and where the remainder term
- <math>R_{\alpha}(x, T) \ll x^{-\alpha} + \frac{x^{1-\alpha} \log(x)}{T} + \frac{x^{1-\alpha}}{T^{1-\varepsilon} \log(x)}, </math>
which of course tends to 0 as <math>T \rightarrow \infty</math>. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since <math>\zeta(1/2) < 0</math> we have another similarity in the form of <math>L_{\alpha}(x)</math> to <math>M(x)</math> insomuch as the dominant leading term in the previous formulas predicts a negative bias in the values of these functions over the positive natural numbers x.