The Distribution of Weighted Sums of the Liouville Function and P\'olya's Conjecture

Abstract

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 $x \geq 17$.