Humphries, P; (2013) The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture. Journal of Number Theory , 133 (2) pp. 545-582. 10.1016/j.jnt.2012.08.011.

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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, Lα(x)=∑n⩽xλ(n)/nα, 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 Pólyaʼs 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.

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