On the Liouville function at polynomial arguments (Q6582304)

Let \(\lambda(n)=(-1)^{\Omega(n)}\) be the Liouville function of \(n\), where \(\Omega(n)\) is the total number of prime factors of \(n\). A conjecture of \textit{J. Cassaigne} et al. [Acta Arith. 87, No. 4, 367--390 (1999; Zbl 0920.11053)] asserts that if \(P(x)\) is a nonconstant polynomial with integer coefficients not of the form \(cQ(x)^2\) for some \(c\in {\mathbb Z}\) and \(Q(x)\in {\mathbb Z}[x]\), then for any \(v\in \{\pm 1\}\) the equation \(\lambda(P(n))=v\) has infinitely many positive solutions \(n\). In this paper, the author makes major contributions to this and other related problems. For example, he gives an affirmative answer to the above conjecture if\N\begin{itemize}\N\item [(i)] \(P(x)\) factors in linear factors over \({\mathbb Q}\) (Corollary 2.1). In fact, in this case for each \(v\in \{\pm 1\}\) the equality \(\lambda(P(n))=v\) holds for a positive proportion of \(n\le x\);\N\item[(ii)] \(P(x)=x(x^2-Bx+C)\), where \(B\ge 0\) (Theorem 2.2). The number of such \(n\le x\) is \(\gg_P {\sqrt{x}}\).\N\item[(iii)] \(P(x)=((x+h_1)^2+1)((x+h_2)^2+1)\cdots ((x+h_k)^2+1)\), where \(h_1,\ldots,h_k\) are distinct integers.\N\end{itemize}\NThe author's results are much more general and they cover the situation where the function \(\lambda\) is replaced by some multiplicative function with values in roots of unity and finite range. The author's result also address smooth values of polynomials. Let \(P(x)\in {\mathbb Z}[x]\). Let us say that \(P\) has property \(S\) if for every positive integers \(b,~q\), the proportion of \(n\le x\) in the arithmetic progression \(n\equiv b\pmod q\) such that \(P(n)\) is \(n\)-smooth (the largest prime factor of \(P(n)\) is at most \(n\)) is at least \(\eta_0\), where \(\eta_0\) is some positive constant not depending on \(b\) or \(q\). Then Proposition 10 shows that if \(P(x)\) is either quadratic or factors in linear factors then it has property \(S\). He proves analogous results for almost all polynomials (ordered by height) and for a couple of multivariate polynomials. For example, Theorem 2.12 shows that for all integers \(1\le a,b\le 100\), and for all \(v\in \{\pm 1\}\), each of the equations \(\lambda(am^3+bn^4)=v\) and \(\lambda(am^2+bn^k)=v\) (any fixed \(k\ge 1\)) have infinitely many coprime solutions \((m,n)\).\N\NThe proofs proceed in two cases according to whether \(g\) is pretentious in the sense of Granville and Soundararajan (at finite distance from some Dirichlet character), or not and uses a lot of machinery from analytic and algebraic number theory such as Fourier analysis, negative Pell equations, as well as explicit calculations.