How to get values of Summatory Liouville function from Mertens function? All:
For Liouville function λ(n), we can define summatory Liouville as the accumulated sum of of λ(n). Mertens function is the accumulated sum of Mobius function.
Is there any ways to get the value summatory Liouville function from Mertens function ?
The reason I asked this question is that, it seems there are some efficient algorithms to calculate Mertens function. I just want to know how to adopt those similar algorithms to calculate summatory Liouville function.
My personal goal is to find the counter example of  Pólya conjecture  as in:
https://en.wikipedia.org/wiki/Liouville_function
by calculating summatory Liouville for some larger number.
 A: One can show that
\[\sum_{d \mid n} \lambda(d) = \begin{cases} 1 & \text{if $d$ is a perfect square,} \\ 0 & \text{otherwise.} \end{cases}\]
Indeed, $\lambda(n)$ is multiplicative, so one only needs to check this on prime powers, and this is easy.
Via Möbius inversion, it therefore follows that
\[\lambda(n) = \sum_{d^2 m = n} \mu(m),\]
and so
\[L(x) = \sum_{n \leq x} \lambda(n) = \sum_{d^2 m \leq x} \mu(m) = \sum_{d \leq \sqrt{x}} \sum_{m \leq x/d^2} \mu(m) = \sum_{d \leq \sqrt{x}} M\left(\frac{x}{d^2}\right).\]
This is the same answer that user1952009 got, but directly using Möbius inversion instead of Mellin transforms and Dirichlet series.
(Note that these are essentially the same, however; one can think of Dirichlet series as being the generating function of multiplicative functions, and multiplication of Dirichlet series is the same as Dirichlet convolution. Mellin transforms then relate Dirichlet series to summatory functions.)
Similarly, one can show that
\[\mu(n) = \sum_{d^2 m = n} \mu(d) \lambda(m),\]
and so
\[M(x) = \sum_{n \leq x} \mu(n) = \sum_{d^2 m \leq x} \mu(d) \lambda(m) = \sum_{d \leq \sqrt{x}} \mu(d) \sum_{m \leq x/d^2} \lambda(m) = \sum_{d \leq \sqrt{x}} \mu(d) L\left(\frac{x}{d^2}\right).\]
All this being said, there are faster ways to compute the summatory function of the Liouville function. For example, this paper of R. Sherman Lehman gives more complicated recursive formulae that allowed him to compute values of $L(x)$ reasonably quickly by the standards of 1960. More recently, work of Borwein, Ferguson, and Mossinghoff and Mossinghoff and Trudgian involve even faster algorithms to compute $L(x)$.
