  Under the assumption of the Riemann Hypothesis, the Linear Independence
Hypothesis, and a bound on negative discrete moments of the Riemann zeta
function, we prove the existence of a limiting logarithmic distribution of the
normalisation of the weighted sum of the Liouville function, $L_{\alpha}(x) =
\sum_{n \leq x}{\lambda(n) / n^{\alpha}}$, for $0 \leq \alpha < 1/2$. Using
this, we conditionally show that these weighted sums have a negative bias, but
that for each $0 \leq \alpha < 1/2$, the set of all $x \geq 1$ for which
$L_{\alpha}(x)$ is positive has positive logarithmic density. For $\alpha = 0$,
this gives a conditional proof that the set of counterexamples to P\'olya's
conjecture has positive logarithmic density. Finally, when $\alpha = 1/2$, we
conditionally prove that $L_{\alpha}(x)$ is negative outside a set of
logarithmic density zero, thereby lending support to a conjecture of
Mossinghoff and Trudgian that this weighted sum is nonpositive for all $x \geq
17$.
