The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field (Q2922408)

The objective of this paper is to prove a function field version of Chowla's conjecture. In the number field case we have that the Riemann Hypothesis is equivalent to \(\sum_{n\leq N} \mu(n)= O(N^{1/2+o(1)})\) where \(\mu\) is the usual Möbius function. \textit{S. Chowla}'s conjecture [The Riemann hypothesis and Hilbert's tenth problem. (Mathematics and its Applications. Vol. 4). Glasgow: Blackie \& Son Ltd (1965; Zbl 0133.30003)] on the auto-correlation of the Möbius function asserts that given an \(r\)--tuple of distinct integers \(\alpha_1,\ldots,\alpha_r\) and \(\varepsilon_i \in \{1,2\}\), not all even, it holds that NEWLINE\[NEWLINE \lim_{N\to\infty}\frac{1}{N}\sum_{n\leq N}\mu(n+\alpha_1)^{ \varepsilon_1}\cdots \mu(n+\alpha_r)^{\varepsilon_r}=0. NEWLINE\]NEWLINE The case \(r=1\), the only known case, is equivalent with the prime number theorem.NEWLINENEWLINEIn the function field case, we consider \({\mathbb F}_q\) the finite field of \(q\) elements. Let \(R_x={\mathbb F}_q[x]\) be the ring of polynomials over \({\mathbb F}_q\). The Möbius function of a non--zero polynomial \(F\in R_x\) is defined to be \(\mu(F)= (-1)^r\) if \(F=cP_1\cdots P_r\) with \(c\in {\mathbb F}_q\setminus \{0\}\) and \(P_1,\ldots, P_r\in R_x\) distinct monic irreducible polynomials, and \(\mu(F)=0\) otherwise.NEWLINENEWLINELet \(M_n\) be the subset of \(R_x\) of monic polynomials of degree \(n\), \(\#M_n=q^n\). It is well known that the number of square--free polynomials in \(M_n\) is \(q^n-q^{n-1}\), for \(n>1\). Hence, given \(r\) distinct polynomials \(\alpha_1,\ldots,\alpha_r\in R_x\), with \(\deg \alpha_i<n\), the number of \(F\in M_n\) for which of \(F(x)+\alpha_j (x)\) are square--free is \(q^n+O(rq^{n-1})\) as \(q\to \infty\). Set NEWLINE\[NEWLINE C(\alpha_1,\ldots,\alpha_r; n)=\sum_{F\in M_n} \mu(F+\alpha_1)^{ \varepsilon_1}\cdots \mu(F+\alpha_r)^{\varepsilon_r} NEWLINE\]NEWLINE with \(\varepsilon_i\in\{1,2\}\) not all even and \(\alpha_1,\ldots, \alpha_r \in R_x\) distinct polynomials with \(\deg \alpha_j<n\). Then \(C(\alpha_1;n)=0\) for \(n>1\) and \(C(\alpha_1,\ldots,\alpha_r; 1)=(-1)^{\varepsilon}q^n\) where \(\varepsilon=\sum_{i=1}^r \varepsilon_i\).NEWLINENEWLINEThe main result of the paper is the following analogue of Chowla's conjecture. For fixed \(r>1\), \(n>1\) and \(q\) odd and for any choice of distinct \(\alpha_1,\ldots, \alpha_r\in R_x\) with \(\deg \alpha_j<n\) and \(\varepsilon_j\in\{1,2\}\), not all even, we have NEWLINE\[NEWLINE |C(\alpha_1,\ldots,\alpha_r;n)|\leq 2rnq^{n-1/2}+3rn^2 q^{n-1}. NEWLINE\]NEWLINE In particular, for fixed \(n>1\), NEWLINE\[NEWLINE \lim_{q\to\infty}\frac{1}{q^n}\sum_{F\in M_n} \mu(F+\alpha_1)^{ \varepsilon_1}\cdots \mu(F+\alpha_r)^{\varepsilon_r}=0. NEWLINE\]NEWLINE The starting point for the proof is Pellet's formula \(\mu(F)=(-1)^{ \deg F}\chi_2(\roman{disc}(F))\) where \(q\) is odd and \(\chi_2\) is the quadratic character of \({\mathbb F}_q\).