Index admissible operator, 131 admissible set of operators, 142 admissible set of symbols, 26 Bernoulli numbers, 79 Bernoulli polynomials, 80 Beta function, 3 bidegree of conformal covariance, 174 canonical discrete sum, 94 canonical integral, 28 canonical trace, 123 classical symbol on an open set, 106 closed linear forms, 22 conformal anomaly, 180 conformal Laplacian, 175 conformality, 174 conformally covariant operators, 174 conformally invariant functional, 177 continuity of a weighted trace, 148 continuity of the residue, 122 convex (closed) polytope, 83 covariant linear forms on symbols, 32 covariant set of symbols, 32 cut-off Gamma function, 11 cut-off integral, 36, 86 cut-off sum, 82, 89 defect formula, 148 dimensional regularisation, 72 discrepancies, 12, 13, 39, 74, 139 elliptic operator, 111 energy-momentum tensor, 181 Epstein’s ζ-function, 102 Euler’s constant, 2 Euler’s lemma, 21 Euler-Maclaurin formula, 80, 81 excision function around the origin, 17 expanded polytope, 84 formal classical symbols, 19 Fr´ echet topology on classical operators, 117 Fr´ echet topology on classical symbols, 19 Gamma function, 2 Hadamard integral and method, 7, 36 Hadamard extension of Gamma, 11 Hadamard finite part method, 1, 7 heat-kernel regularised trace, 158 holomorphic regularisation on operators, 135 holomorphic family in a Banach space, 56 holomorphic family of operators, 56, 133 holomorphic family of symbols, 57 holomorphic regularisation on symbols, 70 homogeneous distribution, 2, 16 Hurwitz zeta function, 98 index of D+, 159 integral polytopes, 83 kernel of an operator, 108 Khovanskii-Pukhlikov formula, 84 Laplace-Beltrami operator, 162, 175 leading symbol, 18, 107, 111, 116 Lichnerowicz formula, 162 local conformal anomaly, 181 local discrepancies, 139 localisation subordinated to a chart, 114 locality of the noncommutative residue, 119 log-polyhomogeneous operator, 111 log-polyhomogeneous order, 107 log-polyhomogeneous symbol, 17, 107 log-type, 17, 107 minimal subtraction scheme, 54 modified dimensional regularisation, 70 multiplicative anomaly of a weighted determinant, 170 multiplicative anomaly of the zeta determinant, 173 noncommutative residue of a symbol, 21 noncommutative residue of an operator, 121 nonlocality of canonical traces, 123 nonlocality of regularised traces, 136 189

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