The distribution of weighted sums of the Liouville function and Pólyaʼs conjecture

@inproceedings{Humphries2013TheDO,
  title={The distribution of weighted sums of the Liouville function and P{\'o}lyaʼs conjecture},
  author={Peter Humphries},
  year={2013}
}
  • Peter Humphries
  • Published 2013
  • Mathematics
  • 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… CONTINUE READING

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