{
"id": "2010.07924",
"version": "v1",
"published": "2020-10-15T17:55:28.000Z",
"updated": "2020-10-15T17:55:28.000Z",
"title": "On the Liouville function at polynomial arguments",
"authors": [
"Joni Teräväinen"
],
"comment": "43 pages",
"categories": [
"math.NT"
],
"abstract": "Let $\\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-S\\'ark\\\"ozy asks to show that if $P(x)\\in \\mathbb{Z}[x]$, then the sequence $\\lambda(P(n))$ changes sign infinitely often, assuming only that $P(x)$ is not the square of another polynomial. We show that the sequence $\\lambda(P(n))$ indeed changes sign infinitely often, provided that either (i) $P$ factorizes into linear factors over the rationals; or (ii) $P$ is a reducible cubic polynomial; or (iii) $P$ factorizes into a product of any number of quadratics of a certain type; or (iv) $P$ is any polynomial not belonging to an exceptional set of density zero. Concerning (i), we prove more generally that the partial sums of $g(P(n))$ for $g$ a bounded multiplicative function exhibit nontrivial cancellation under necessary and sufficient conditions on $g$. This establishes a \"99% version\" of Elliott's conjecture for multiplicative functions taking values in the roots of unity of some order. Part (iv) also generalizes to the setting of $g(P(n))$ and provides a multiplicative function analogue of a recent result of Skorobogatov and Sofos on almost all polynomials attaining a prime value.",
"revisions": [
{
"version": "v1",
"updated": "2020-10-15T17:55:28.000Z"
}
],
"analyses": {
"subjects": [
"11N37",
"11B30"
],
"keywords": [
"liouville function",
"polynomial arguments",
"changes sign",
"multiplicative function analogue",
"elliotts conjecture"
],
"note": {
"typesetting": "TeX",
"pages": 43,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}