Determinants of Multiplicative Toeplitz Matrices1

In this paper we study matrices A = (aij) whose (i, j) -entry is a function of i/j; that is, aij = f(i/j) for some f : Q → C. We obtain a formula for the truncated determinants in the case where f is multiplicative, linking them to determinants of truncated Toeplitz matrices. We apply our formula to obtain several determinants of number-theoretic matrices. Mathematics Subject Classification 2000. Primary: 11C20. Secondary: 15A15, 15A23, 65F40. Introduction In this article, we consider matrices and their determinants of the form   c(1) c( 1 2 ) c( 1 3 ) c( 1 4 ) · · · c(2) c(1) c( 23 ) c( 1 2 ) · · · c(3) c( 32 ) c(1) c( 3 4 ) · · · c(4) c(2) c( 43 ) c(1) · · · .. .. .. .. . . .   , (0.1) where the (i, j)-entry is a function of i/j. They are characterised by being constant along lines where i/j is a given positive rational. In this sense they resemble Toeplitz matrices, which are constant on lines parallel to the diagonal; i.e. where i − j is a given integer. For this reason we shall call them multiplicative Toeplitz matrices. These matrices represent linear operators between various spaces. To identify these spaces, and to determine when they are bounded are interesting questions in themselves, but for the purposes of this paper, these will not concern us. We shall be mainly concerned with determinants of truncated matrices. Toeplitz matrices are most usefully studied by associating them with a function (or ‘symbol’) whose Fourier coefficients make up the matrix. Indeed, they generate bounded operators on ` if and only if the matrix entries are the Fourier coefficients of an essentially bounded function on T. For matrices of the form (0.1), we associate, by analogy, the series ∑ q∈Q+ c(q)q, (0.2) where q ranges over the positive rationals. We shall first make sense of such series in §1. In §2 , we concern ourselves with the case when c(·) is a multiplicative function on the positive rationals. Then the matrix (0.1) and the series (0.2) are shown to factorise as ‘Euler products’. In §3, we consider the determinants of truncated multiplicative Toeplitz matrices for which c(·) is multiplicative. The ‘Euler product’ formula ceases to hold when we truncate the matrices, but we show in Theorem 3.2 that it is miraculously recovered on taking determinants. This reduces the problem of evaluating such determinants to those whose non-zero entries lie on ‘lines’ i/j = p for p prime and k ∈ Z. It is shown in Theorem 3.1 that these can be evaluated in terms of determinants of Toeplitz matrices. The proofs are done separately in §4. Using this formula, and 1In Acta Arithmetica 125 (2006) 265-284.