The Euler-Maclaurin formula and Birkhoff-Hopf factorisation: discretisation and quantisation Extension: Enhanced discrete sums; conical and branched zeta values

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The proposed programme extension studies the connections between "enhanced" discrete sums, and more specifically multiple zeta values and their renormalisation, the enhancing being of two types,-on the combinatorial side, branched zeta functions associated with rooted trees, with ladder trees corresponding to multiple zeta functions;- on the geometric side, conical zeta functions associated with convex cones, with Chen cones corresponding to multiple zeta functions. The branching procedure inherent to the tree structure is transposed to families of cones, leading to branched cones and meromorphic germs arising from branched zeta functions are studied with the conical structure inherent to the pole structure.