Cassaigne et al.'s sign-change conjecture for the Liouville function

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Let f(x)∈Z[x]f(x)\in\mathbb{Z}[x]. Suppose that there are no integer bb and g(x)∈Z[x]g(x)\in\mathbb{Z}[x] such that

f(x)=b(g(x))2.f(x)=b(g(x))^2.

Cassaigne et al.'s conjecture. The values λ(f(n))\lambda(f(n)) change sign infinitely often.

This conjecture concerns sign changes of the Liouville function along polynomial sequences. The paper's abstract states that the special case f(n)=n2+df(n)=n^2+d, for every non-zero integer dd, is proved, while the general conjecture remains open.

References

Primary source

Anitha Srinivasan, “Infinitely many sign changes of the Liouville function on x^2+d”, arXiv:2104.15004 (2021).

Additional references

3 papers in this index state this conjecture (2011–2021). The statement above is taken from the most recent of them; the others are arXiv:1202.0471, arXiv:1109.3107.

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