The Distribution of Weighted Sums of the Liouville Function and Pólya’s Conjecture

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, Lα(x) = ∑ n≤x λ(n)/n α, for 0 ≤ α < 1/2. Using this, we conditionally show that these weighted sums have a negative bias, but that for each 0 ≤ α < 1/2, the set of all x ≥ 1 for which Lα(x) is positive has positive logarithmic density. For α = 0, this gives a conditional proof that the set of counterexamples to Pólya’s conjecture has positive logarithmic density. Finally, when α = 1/2, we conditionally prove that Lα(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 ≥ 17.