Dirichlet matrices: determinants, permanents and the \textit{Factorisatio Numerorum} problem
A function \(g:\mathbb{N}\to\mathbb{C}\) and a vector \(\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{C}^n\) define the \(n\times n\) Dirichlet matrix \(D(n,\alpha,g)=(d_{ij})\), where \(d_{1j}=\alpha_j\), \(d_{ij}=g(i/j)\) if \(i>1\) and \(j\mid i\), and \(d_{ij}=0\) otherwise. The author proves that if \(g(1)\ne 0\), then \[ \det{D(n,\alpha,g)}=g(1)^{n-1}\Big(\alpha_1- \sum_{j=2}^ng(j)\frac{\det{D(n_j,\alpha^{(j)},g)}}{g(1)^{n_j}}\Big), \] where \[ n_j=\lfloor\frac{n}{j}\rfloor,\quad\alpha^{(j)}=(\alpha_j,\alpha_{2j},\dots,\alpha_{n_jj}). \] By showing that \(\mathrm{per}\,D(n,\alpha,g)=-\det{D(n,-\alpha,\tilde{g})}\), where \(\tilde{g}(1)=g(1)\) and \(\tilde{g}(j)=-g(j)\) for \(j>1\), he obtains the corresponding formula for the permanent. Let \(1\) denote any identically-one function and also any vector \((1,\dots,1)\). \textit{H. S. Wilf} [Electron J. Combin. 11, R10 (2004; Zbl 1077.15009)] proved that \[ \mathrm{per}\,D(n,1,1)=F(n)=\sum_{j=1}^nf(j), \] where \(f(1)=1\) and \(f(j)\) is the number of ordered factorizations of \(j\ge 2\). \textit{L. Kalmár} [Acta Szeged Sci. Math. 5, 95--107 (1931; JFM 57.1365.02)] proved that \[ \lim_{n\to\infty}\frac{F(n)}{n^\rho}=-\frac{1}{\rho\zeta'(\rho)}, \] where \(\zeta\) is the Riemann zeta function and \(\zeta(\rho)=2\). More generally, the present author studies the asymptotics of \[ F(n,\nu,g)=\mathrm{per}\,D(n,\alpha[n,\nu],g), \] where \(\nu\in\mathbb{R}\), \[ \alpha[n,\nu]=\Big(1,\frac{1}{2^\nu},\dots,\frac{1}{n^\nu}\Big), \] and the values of \(g\) are real and nonnegative.
scientific article; zbMATH DE number 7390152
scientific article