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

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

2011-08-07

arxiv.org/abs/1108.1524v1

Mathematics

Number Theory

32 pages

Scientific paper

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 $x \geq 17$.

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