Finite products of regularized products
The regularized product \(\widehat {\prod}_m a_m\) of a countable set \(\{a_1,a_2,\dots\}\) of non-zero complex numbers is defined by \(\widehat {\prod}_m a_m =\exp(-f'(0)), \) where it is assumed that \(f(s)=\sum_{m} a_m^{-s}\) converges in some right half-plane and has a meromorphic continuation to the \(s\)-plane which is regular at \(s=0\). In general, the product \((\widehat {\prod}_m a_m)\cdot(\widehat {\prod}_m b_m)\) of two regularized products is not equal to the regularized product \(\widehat {\prod}_m(a_m \cdot b_m)\). In the paper, the authors investigate the discrepancy \(F\), defined by \[ \exp(F)=\frac{\widehat {\prod}_m(a_m \cdot b_m)}{(\widehat {\prod}_m a_m)\cdot(\widehat {\prod}_mb_m)}. \] The main result of the notices is stated in terms of Dirichlet series rather than regularized products. Let \(P_j(x_1,\dots,x_r) \in \mathbb C[x_1,\dots,x_r]\) be \(n\) complex polynomials in \(r\) variables, all satisfying Mahler's hypothesis on \(P\) (that \(P(x_1,\dots,x_r) \in \mathbb C[x_1,\dots,x_r]\) does not vanish anywhere in the closed real first octant \(x_i \geq 0\), \(i=1,\dots,r\); its homogeneous part of highest degree is not constant and vanishes nowhere in the closed real first octant, except \(P(0,\dots,0)=0\)), and let \(\varphi(x_1,\dots,x_r) \in \mathbb C[x_1,\dots,x_r]\) be any polynomial in \((x_1,\dots,x_r)\). Define, for \(\Re s \gg 0\), \( f_j(s)=\sum_{(m_1,\dots,m_r)\in \mathbb N^r}\frac{\varphi(m_1,\ldots,m_r)}{P_j(m_1,\dots,m_r)^s} \), and \( f_0(s)=\sum_{(m_1,\dots,m_r)\in \mathbb N^r}\frac{\varphi(m_1,\dots,m_r)}{\prod_{j=1}^{n}P_j(m_1,\dots,m_r)^s} \), \(j=1,\dots,n\), and, after analytic continuation the \(f_j\), \(j=0,\dots,n\), \[ F=-f_0'(0)+\sum_{j=1}^{n}f_j'(0). \] Then \(F\) is a polynomial of degree at most \(r+\deg(\varphi)\) in the coefficients of the \(P_j\) of non-maximal degree.
scientific article; zbMATH DE number 5302011