The multiplicative Hilbert matrix

Abstract

It is observed that the infinite matrix with entries (mnlog⁡(mn))−1 for m,n≥2 appears as the matrix of the integral operator Hf(s):=∫1/2+∞f(w)(ζ(w+s)−1)dw with respect to the basis (n−s)n≥2; here ζ(s) is the Riemann zeta function and H is defined on the Hilbert space H02 of Dirichlet series vanishing at +∞ 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 T∞. 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 π and that it has a purely continuous spectrum which is the interval [0,π]; these results are in agreement with known facts about the classical Hilbert matrix. It is shown that the matrix (m1/pn(p−1)/plog⁡(mn))−1 has norm π/sin⁡(π/p) when acting on ℓp for 1<p<∞. However, the multiplicative Hilbert matrix fails to define a bounded operator on H0p for p≠2, where H0p are Hp spaces of Dirichlet series. It remains an interesting problem to decide whether the analytic symbol ∑n≥2(log⁡n)−1n−s−1/2 of the multiplicative Hilbert matrix arises as the Riesz projection of a bounded function on the infinite-dimensional torus T∞.