DISCREPANCIES OF PRODUCTS OF ZETA-REGULARIZED PRODUCTS
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Date
2012
Journal Title
Journal ISSN
Volume Title
Publisher
INT PRESS BOSTON, INC
Abstract
Zeta-regularized products (Pi) over cap (m) a(m) are known not to commute with finite products, so one studies the discrepancy F-n given by
exp(F-n) := (Pi) over cap (m) (Pi(n)(j=1) a(m,j))/Pi(n)(j=1) ((Pi) over cap (m)a(m,j)).
For a rather general class of products, associated to polynomials P-j in several variables, we show that the discrepancy F-n(P-1, ... ,P-n) of n products is a sum of pairwise contributions F-2(P-i, P-j). Namely,
(Sigma(n)(j=1) deg P-j) F-n(P-1, ... ,P-n) = Sigma(1<i< j<n) (deg P-i + deg P-j) F-2(P-i, P-j).
Thus, there are no higher interactions behind the non-commutativity.
exp(F-n) := (Pi) over cap (m) (Pi(n)(j=1) a(m,j))/Pi(n)(j=1) ((Pi) over cap (m)a(m,j)).
For a rather general class of products, associated to polynomials P-j in several variables, we show that the discrepancy F-n(P-1, ... ,P-n) of n products is a sum of pairwise contributions F-2(P-i, P-j). Namely,
(Sigma(n)(j=1) deg P-j) F-n(P-1, ... ,P-n) = Sigma(1<i< j<n) (deg P-i + deg P-j) F-2(P-i, P-j).
Thus, there are no higher interactions behind the non-commutativity.
Description
Keywords
Regularized products, Dirichlet series, DIRICHLET SERIES, POLYNOMIALS