Liouville function

The Liouville lambda function, denoted by $λ(n)$ and named after Joseph Liouville, is an important arithmetic function. Its value is $+1$ if $n$ is the product of an even number of prime numbers, and $−1$ if it is the product of an odd number of primes.

Explicitly, the fundamental theorem of arithmetic states that any positive integer $n$ can be represented uniquely as a product of powers of primes: $n = p_{1}^{a_{1}} &ctdot; p_{k}^{a_{k}}|undefined$, where $p_{1} < p_{2} < ... < p_{k}$ are primes and the $a_{j}$ are positive integers. ($1$ is given by the empty product.) The prime omega functions count the number of primes,  with ($Ω$) or without ($ω$) multiplicity:


 * $$ \omega(n) = k, $$
 * $$ \Omega(n) = a_1 + a_2 + \cdots + a_k. $$

$λ(n)$ is defined by the formula


 * $$ \lambda(n) = (-1)^{\Omega(n)} $$

.

$λ$ is completely multiplicative since $Ω(n)$ is completely additive, i.e.: $Ω(ab) = Ω(a) + Ω(b)$. Since $1$ has no prime factors, $Ω(1) = 0$, so $λ(1) = 1$.

It is related to the Möbius function $μ(n)$. Write $n$ as $n = a^{2}b$, where $b$ is squarefree, i.e., $ω(b) = Ω(b)$. Then


 * $$ \lambda(n) = \mu(b). $$

The sum of the Liouville function over the divisors of $n$ is the characteristic function of the squares:



\sum_{d|n}\lambda(d) = \begin{cases} 1 & \text{if }n\text{ is a perfect square,} \\ 0 & \text{otherwise.} \end{cases} $$

Möbius inversion of this formula yields
 * $$\lambda(n) = \sum_{d^2|n} \mu\left(\frac{n}{d^2}\right).$$

The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, $&lambda;^{–1}(n) = &vert;&mu;(n)&vert; = &mu;^{2}(n)$, the characteristic function of the squarefree integers. We also have that $&lambda;(n) = &mu;^{2}(n)$.

Series
The Dirichlet series for the Liouville function is related to the Riemann zeta function by


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

Also:


 * $$\sum\limits_{n=1}^{\infty} \frac{\lambda(n)\ln n}{n}=-\zeta(2)=-\frac{\pi^2}{6}.$$

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.

Conjectures on weighted summatory functions


The Pólya problem is a question raised made by George Pólya in 1919. Defining


 * $$L(n) = \sum_{k=1}^n \lambda(k)$$ ,

the problem asks whether $$L(n)\leq 0$$ for 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$\sqrt{n}$ for infinitely many positive integers n, while it can also be shown via the same methods that L(n) < -1.3892783$\sqrt{n}$ for infinitely many positive integers n.

For any $$\varepsilon > 0$$, assuming the Riemann hypothesis, we have that the summatory function $$L(x) \equiv L_0(x)$$ is bounded by


 * $$L(x) = O\left(\sqrt{x} \exp\left(C \cdot \log^{1/2}(x) \left(\log\log x\right)^{5/2+\varepsilon}\right)\right),$$

where the $$C > 0$$ is some absolute limiting constant.

Define the related sum


 * $$T(n) = \sum_{k=1}^n \frac{\lambda(k)}{k}.$$

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, 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 $$\alpha \in \mathbb{R}$$ as follows for positive integers x where (as above) we have the special cases $$L(x) := L_0(x)$$ and $$T(x) = L_1(x)$$


 * $$L_{\alpha}(x) := \sum_{n \leq x} \frac{\lambda(n)}{n^{\alpha}}.$$

These $$\alpha^{-1}$$-weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Moebius function. In fact, we have that the so-termed non-weighted, or ordinary function $$L(x)$$ precisely corresponds to the sum


 * $$L(x) = \sum_{d^2 \leq x} M\left(\frac{x}{d^2}\right) = \sum_{d^2 \leq x} \sum_{n \leq \frac{x}{d^2}} \mu(n).$$

Moreover, these functions satisfy similar bounding asymptotic relations. For example, whenever $$0 \leq \alpha \leq \frac{1}{2}$$, we see that there exists an absolute constant $$C_{\alpha} > 0$$ such that


 * $$L_{\alpha}(x) = O\left(x^{1-\alpha}\exp\left(-C_{\alpha} \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right).$$

By an application of Perron's formula, or equivalently by a key (inverse) Mellin transform, we have that


 * $$\frac{\zeta(2\alpha+2s)}{\zeta(\alpha+s)} = s \cdot \int_1^{\infty} \frac{L_{\alpha}(x)}{x^{s+1}} dx,$$

which then can be inverted via the inverse transform to show that for $$x > 1$$, $$T \geq 1$$ and $$0 \leq \alpha < \frac{1}{2}$$


 * $$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), $$

where we can take $$\sigma_0 := 1-\alpha+1 / \log(x)$$, and with the remainder terms defined such that $$E_{\alpha}(x) = O(x^{-\alpha})$$ and $$R_{\alpha}(x, T) \rightarrow 0$$ as $$T \rightarrow \infty$$.

In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by $$\rho = \frac{1}{2} + \imath\gamma$$, of the Riemann zeta function are simple, then for any $$0 \leq \alpha < \frac{1}{2}$$ and $$ x \geq 1$$ there exists an infinite sequence of $$\{T_v\}_{v \geq 1}$$ which satisfies that $$v \leq T_v \leq v+1$$ for all v such that


 * $$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), $$

where for any increasingly small $$0 < \varepsilon < \frac{1}{2}-\alpha$$ we define


 * $$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,$$

and where the remainder term


 * $$R_{\alpha}(x, T) \ll x^{-\alpha} + \frac{x^{1-\alpha} \log(x)}{T} + \frac{x^{1-\alpha}}{T^{1-\varepsilon} \log(x)}, $$

which of course tends to 0 as $$T \rightarrow \infty$$. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since $$\zeta(1/2) < 0$$ we have another similarity in the form of $$L_{\alpha}(x)$$ to $$M(x)$$ in so much 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.