The Fredholm determinant of an almost periodic arithmetical function (Q5943763)

Denote by \({\mathcal D}\) the \(\mathbb{C}\)-vector space of linear combinations of exponential sums \(e_\alpha\), \(\alpha\in \mathbb{Q}\), and by \({\mathcal D}^2\) the closure of \({\mathcal D}\) with respect to the (semi-)norm \[ \|f\|_2= \Biggl( \limsup_{N\to\infty} \frac{1}{2N+1} \sum_{|n|\leq N}|f(n)|^2 \Biggr)^{\frac 12}. \] (\({\mathcal D}^2\) is called the space of 2-limit periodic functions.) The Fredholm determinant \(D(f;z)\) of \(f\in{\mathcal D}^2\) is defined as an everywhere convergent power series whose coefficients are mean values of certain determinants. The author derives the Weierstraß product representation \[ D(f;z)= e^{-f(0)z} \prod_{\alpha\in \mathbb{Q}/\mathbb{Z}} (1- \widehat{f}(\alpha)z) e^{\widehat{f}(\alpha)z}. \] In particular this shows that the zeros of \(D(f;z)\) are the reciprocals of the nonvanishing Fourier coefficients \(\widehat{f}(\alpha)\) of \(f\).