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Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\\frac{\\det_\\zeta\\Big(\\prod_{i=1}^n A_i\\Big)}{\\prod_{i=1}^n \\det_\\zeta(A_i)} attached to n (suitable) operators A_1,...,A_n. We show that if the A_i are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely M_n(A_1,...,A_n)^{m_1+...+m_n} =\\prod_{1\\le i<j\\le n}M_2(A_i,A_j)^{m_i+m_j}, where m_j is the order of A_j. 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Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\\frac{\\det_\\zeta\\Big(\\prod_{i=1}^n A_i\\Big)}{\\prod_{i=1}^n \\det_\\zeta(A_i)} attached to n (suitable) operators A_1,...,A_n. We show that if the A_i are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely M_n(A_1,...,A_n)^{m_1+...+m_n} =\\prod_{1\\le i<j\\le n}M_2(A_i,A_j)^{m_i+m_j}, where m_j is the order of A_j. 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