Ramanujan coefficients and Hankel matrices (Q1924909)

Every arithmetic function \(f: \mathbb{N}\to \mathbb{C}\) has Hankel determinants \[ h_r:= \text{det} \begin{pmatrix} f(1) & f(2) &\ldots & f(r)\\ f(2) & f(3) &\ldots & f(r+1)\\ \vdots &&& \vdots\\ f(r) & f(r+ 1) &\ldots & f(2r- 1) \end{pmatrix}, \qquad r\in \mathbb{N}, \] and every almost-even function \(f\) has Ramanujan-coefficients \(a_r:= (1/ \varphi (r)) M(fc_r)\), \(r\in \mathbb{N}\). We prove: (i) every arithmetic function \(f\) with \(h_r= 0\) \(\forall r> r_0\) solves a linear recurrence \[ f(n+ k)+ b_{k-1} f(n+ k-1)+ \dots+ b_0 f(n)= 0, \qquad b_j\in \mathbb{C}, \quad \forall n\geq n_0; \tag \(*\) \] (ii) every solution \(f\) of \((*)\) with \(f(n)= o(n)\) has mean values \(M(fc_r)= 0\), \(\forall r> r_0\). From this we conclude that an almost-even function \(f\) with \(a_r \neq 0\) for infinitely many \(r\) has Hankel determinants \(h_r\neq 0\) for infinitely many \(r\). This generalizes a theorem of \textit{W. Schwarz} [Banach Cent. Publ. 17, 463-498 (1985; Zbl 0598.10049)].