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}\cdots p_k^{a_k}    where  p1 < p2 < ... < pk are primes and the aj 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:
ω(n) = k,
Ω(n) = a1 + a2 + ... + ak.
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 = a2b 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) = |\mu(n)| = \mu^2(n), the characteristic function of the squarefree integers.
We also have that \lambda(n) \mu(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}.
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
thumb|none|Summatory Liouville function L(n) up to n = 104.
The readily visible oscillations are due to the first non-trivial zero of the Riemann zeta function.
thumb|none|Harmonic Summatory Liouville function T(n) up to n = 103
The Pólya conjecture is a conjecture made by George Pólya in 1919.
Defining
L(n) = \sum_{k=1}^n \lambda(k) ,
the conjecture states that L(n)\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 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.
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 Lioville 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.
References
