The Distribution of Weighted Sums of the Liouville Function and Pólya'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, $L_{\alpha }(x)={\sum }_{n\le x}\lambda (n)/n^{\alpha }$, for $0\le \alpha <1/2$. Using this, we conditionally show that these weighted sums have a negative bias, but that for each $0\le \alpha <1/2$, the set of all $x\ge 1$ 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 Pólya'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 $x\ge 17$.

33 pages. Several minor revisions and corrections based on referee comments, and additional references added

References in corpus (1)

• Upper bounds for the moments of zeta prime rho

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• On a Mertens-type conjecture for number fields
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