Chowla's conjecture for polynomial values

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Let f∈Z[x1,…,xh]f\in\mathbb{Z}[x_1,\ldots,x_h] be a primitive squarefree polynomial, let A\mathcal{A} be an arithmetic progression, and let A(X)\mathcal{A}(X) denote its points of norm at most XX. For a nonzero integer n=∏ppνp(n)n=\prod_p p^{\nu_p(n)}, define Liouville's function by λ(n)=(−1)∑pνp(n)\lambda(n)=(-1)^{\sum_p\nu_p(n)}. Chowla's conjecture. For every arithmetic progression A\mathcal{A},

∑v∈A(X)λ(f(v))=o(X2).\sum_{{\rm v}\in\mathcal{A}(X)}\lambda(f({\rm v}))=o(X^2).

The conjecture predicts cancellation in the Liouville values of a polynomial along arithmetic progressions. The source gives no resolution and introduces it as a separate conjectural input.

References

Primary source

Julie Desjardins, “On the variation of the root number in families of elliptic curves”, arXiv:1610.07440 (2018).

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