With $\mu(n)$ being Möbius function and
$M(x) = \sum\_{n \le x} \mu(n)$ being Martens function:

Is it possible to prove that the difference between two diverging terms given below
$$ \lim\_{ x \to \infty} \Big ( \frac{M(x)} {\sqrt{x}} - \sum\_{n \le x} \frac{\mu(n)}{\sqrt{n}} \Big) $$ converges?

Both terms above diverge but, computationally, the difference between them seems to be converging to $-1/\zeta(1/2)$.