Discrepancies of products of zeta-regularized products

Zeta-regularized products $\p a_m$ are known not tocommute with finite products, soone studies the discrepancy $F_n$ given by\vspace*{6pt}\[\exp(F_n):=\frac{\p \left(\prod_{j=1}^na_{m,j}\right)}{\prod_{j=1}^n \left(\p a_{m,j}\right)}.\vspace*{6pt}\]For a rather general class of products, associated to polynomials $P_j$in several variables, we show thatthe discrepancy $F_n(P_1,\dots,P_n)$ of $n$ products is a sum of pairwisecontributions $F_2(P_i,P_j)$. Namely,\vspace*{6pt}\[\left(\sum_{j=1}^n \deg P_j \right)F_n(P_1,\dots,P_n)=\sum_{1\le i<j\le n}(\deg P_i +\deg P_j )F_2(P_i,P_j).\vspace*{6pt}\]Thus, there are no higher interactions behind the non-commutativity.

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Published 2 May 2012