The multiplicative Hilbert matrix

Abstract

It is observed that the infinite matrix with entries $(\sqrt{mn}\log (mn))^{-1}$ for $m, n\ge 2$ appears as the matrix of the integral operator $\mathbf{H}f(s):=\int_{1/2}^{+\infty}f(w)(\zeta(w+s)-1)dw$ with respect to the basis $(n^{-s})_{n\ge 2}$; here $\zeta(s)$ is the Riemann zeta function and $H$ is defined on the Hilbert space ${\mathcal H}^2_0$ of Dirichlet series vanishing at $+\infty$ and with square-summable coefficients. This infinite matrix defines a multiplicative Hankel operator according to Helson's terminology or, alternatively, it can be viewed as a bona fide (small) Hankel operator on the infinite-dimensional torus ${\Bbb T}^{\infty}$. By analogy with the standard integral representation of the classical Hilbert matrix, this matrix is referred to as the multiplicative Hilbert matrix. It is shown that its norm equals $\pi$ and that it has a purely continuous spectrum which is the interval $[0,\pi]$; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix $(m^{1/p} n^{(p-1)/p}\log (mn))^{-1}$ has norm $\pi/\sin(\pi /p)$ when acting on $\ell^p$ for $1<p<\infty$. However, the multiplicative Hilbert matrix fails to define a bounded operator on ${\mathcal H}^p_0$ for $p\neq 2$, where ${\mathcal H}^p_0$ are $H^p$ spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol $\sum_{n\ge 2} (\log n)^{-1} n^{-s-1/2}$ of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus ${\Bbb T}^\infty$.