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

  title={The Distribution of Weighted Sums of the Liouville Function and Pólya’s Conjecture},
  author={Peter Humphries},
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… CONTINUE READING

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